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If $P$ is infinite and $P\cong \textrm{End}(P)$, then the size of $P$ is restricted by the following facts.

Lemma. Assume that $P$ is an infinite poset.

1. $P$ has an infinite antichain, an infinite well-ordered subset, or an infinite inversely well-ordered subset.

2. If $|P|>2^{\lambda}$ for some infinite $\lambda$, then $P$ has an antichain, a well-ordered subset, or an inversely well-ordered subset of size $>\lambda$.

3. If $P$ has an antichain of size at least $\kappa$, then $\textrm{End}(P)$ has an antichain of size at least $2^{\kappa}$.

4. If $P$ has a well-ordered or inversely well-ordered subset of size at least $\kappa$, then $\textrm{End}(P)$ has an antichain of size at least $2^{\kappa}$.

In particular, these facts imply that if $P\cong \textrm{End}(P)$, then $|P|\geq \beth_{\omega}$. If also $|P|>2^{\lambda}$ for some infinite $\lambda$, then $|P|>2^{2^{\lambda}}$.

Sketch of proofs.

Part 1 of the Lemma follows from Ramsey's Theorem.

Part 2 of the Lemma follows from the Erdos-Rado Theorem.

For part 3 of the Lemma, if $P$ is an antichain, then $\textrm{End}(P)$ is an antichain of size $2^{|P|}$, so the claim is clear. Otherwise choose $a<b$ in $P$. Let $A\subseteq P$ be an antichain in $P$ of cardinality $\kappa$. It is possible to find a set $\mathcal X$ of pairwise incomparable subsets of $A$ such that $|\mathcal X|=2^{\kappa}$. For each $U\in \mathcal X$ define $f_U\colon P\to P$ by $f_U(x)=b$ if $x$ is in the filter generated by $U$ in $P$, else $f_U(x)=a$. The set $\{f_U | U\in \mathcal X\}$ is an antichain of size $2^{\kappa}$ in $\textrm{End}(P)$.

For part 4 of the lemma, assume that $W = \{w_{\alpha} : \alpha < \kappa\}$ is an enumeration of an increasing $\kappa$-sequence in $P$. Define a partial function $\sigma\colon P\to P$ by taking $\sigma(x)$ to be the least element $w_{\alpha}$ in $W$ such that $x\not\geq w_{\alpha}$. This $\sigma$ is a monotone total function if $W$ is cofinal in $P$, but only a partial function if there is some $q\in P$ above every $w_{\alpha}$. If such $q$ exists, fix one such, and extend the definition of $W$ to include $w_{\kappa}=q$ and extend the definition of $\sigma$ so that $\sigma(x)=q$ if $x\geq w_{\alpha}$ for all $\alpha<\kappa$. Either way, $W$ is well-ordered, of size $\kappa$, and $\sigma\in \textrm{End}(P)$ maps $P$ into $W$ and $\sigma(P)$ is well-ordered and of size $\kappa$. (Note: $\sigma(P)$ might be a proper subset of $W$.)

For every subset $V\subseteq \sigma(P)$ not containing $w_{\kappa}$ define $g\colon \sigma(P)\to \sigma(P)$ by $g(w_{\alpha})=w_{\alpha}$ if $w_{\alpha}\in \sigma(P)\setminus V$ and $g(w_{\alpha})=w_{\alpha+1}$ if $w_{\alpha}\in V$. If $V_1$ and $V_2$ are incomparable subsets of $\sigma(P)$, then $g_{V_1}$ and $g_{V_2}$ are incomparable elements of $\textrm{End}(P)$. There is a set of $2^{\kappa}$-many pairwise incomparable subsets of the $\kappa$-element set $\sigma(P)\setminus\{w_{\kappa}\}$, so $\textrm{End}(P)$ has an antichain of size at least $2^{\kappa}$. \\

If $P$ is infinite and $P\cong \textrm{End}(P)$, then the size of $P$ is restricted by the following facts.

Lemma. Assume that $P$ is an infinite poset.

1. $P$ has an infinite antichain, an infinite well-ordered subset, or an infinite inversely well-ordered subset.

2. If $|P|>2^{\lambda}$ for some infinite $\lambda$, then $P$ has an antichain, a well-ordered subset, or an inversely well-ordered subset of size $>\lambda$.

3. If $P$ has an antichain of size at least $\kappa\geq\omega$, then $\textrm{End}(P)$ has an antichain of size at least $2^{\kappa}$.

4. If $P$ has a well-ordered or inversely well-ordered subset of size at least $\kappa\geq\omega$, then $\textrm{End}(P)$ has an antichain of size at least $2^{\kappa}$.

In particular, these facts imply that if $P\cong \textrm{End}(P)$, then $|P|\geq \beth_{\omega}$. If also $|P|>2^{\lambda}$ for some infinite $\lambda$, then $|P|>2^{2^{\lambda}}$.

Sketch of proofs.

Part 1 of the Lemma follows from Ramsey's Theorem.

Part 2 of the Lemma follows from the Erdos-Rado Theorem.

For part 3 of the Lemma, if $P$ is an antichain, then $\textrm{End}(P)$ is an antichain of size $2^{|P|}$, so the claim is clear. Otherwise choose $a<b$ in $P$. Let $A\subseteq P$ be an antichain in $P$ of cardinality $\kappa$. It is possible to find a set $\mathcal X$ of pairwise incomparable subsets of $A$ such that $|\mathcal X|=2^{\kappa}$. For each $U\in \mathcal X$ define $f_U\colon P\to P$ by $f_U(x)=b$ if $x$ is in the filter generated by $U$ in $P$, else $f_U(x)=a$. The set $\{f_U | U\in \mathcal X\}$ is an antichain of size $2^{\kappa}$ in $\textrm{End}(P)$.

For part 4 of the lemma, assume that $W = \{w_{\alpha} : \alpha < \kappa\}$ is an enumeration of an increasing $\kappa$-sequence in $P$. Define a partial function $\sigma\colon P\to P$ by taking $\sigma(x)$ to be the least element $w_{\alpha}$ in $W$ such that $x\not\geq w_{\alpha}$. This $\sigma$ is a monotone total function if $W$ is cofinal in $P$, but only a partial function if there is some $q\in P$ above every $w_{\alpha}$. If such $q$ exists, fix one such, and extend the definition of $W$ to include $w_{\kappa}=q$ and extend the definition of $\sigma$ so that $\sigma(x)=q$ if $x\geq w_{\alpha}$ for all $\alpha<\kappa$. Either way, $W$ is well-ordered, of size $\kappa$, and $\sigma\in \textrm{End}(P)$ maps $P$ into $W$ and $\sigma(P)$ is well-ordered and of size $\kappa$. (Note: $\sigma(P)$ might be a proper subset of $W$.)

For every subset $V\subseteq \sigma(P)$ not containing $w_{\kappa}$ define $g\colon \sigma(P)\to \sigma(P)$ by $g(w_{\alpha})=w_{\alpha}$ if $w_{\alpha}\in \sigma(P)\setminus V$ and $g(w_{\alpha})=w_{\alpha+1}$ if $w_{\alpha}\in V$. If $V_1$ and $V_2$ are incomparable subsets of $\sigma(P)$, then $g_{V_1}$ and $g_{V_2}$ are incomparable elements of $\textrm{End}(P)$. There is a set of $2^{\kappa}$-many pairwise incomparable subsets of the $\kappa$-element set $\sigma(P)\setminus\{w_{\kappa}\}$, so $\textrm{End}(P)$ has an antichain of size at least $2^{\kappa}$. \\

If $P$ is infinite and $P\cong \textrm{End}(P)$, then the size of $P$ is restricted by the following facts.

Lemma. Assume that $P$ is an infinite poset.

1. $P$ has an infinite antichain, an infinite well-ordered subset, or an infinite inversely well-ordered subset.

2. If $|P|>2^{\lambda}$ for some infinite $\lambda$, then $P$ has an antichain, a well-ordered subset, or an inversely well-ordered subset of size $>\lambda$.

3. If $P$ has an antichain of size at least $\kappa$, then $\textrm{End}(P)$ has an antichain of size at least $2^{\kappa}$.

4. If $P$ has a well-ordered or inversely well-ordered subset of size at least $\kappa$, then $\textrm{End}(P)$ has an antichain of size at least $2^{\kappa}$.

In particular, these facts imply that if $P\cong \textrm{End}(P)$, then $P\geq \beth_{\omega}$. If also $|P|>2^{\lambda}$ for some infinite $\lambda$, then $|P|>2^{2^{\lambda}}$.

Sketch of proofs.

Part 1 of the Lemma follows from Ramsey's Theorem.

Part 2 of the Lemma follows from the Erdos-Rado Theorem.

For part 3 of the Lemma, if $P$ is an antichain, then $\textrm{End}(P)$ is an antichain of size $2^{|P|}$, so the claim is clear. Otherwise choose $a<b$ in $P$. Let $A\subseteq P$ be an antichain in $P$ of cardinality $\kappa$. It is possible to find a set $\mathcal X$ of pairwise incomparable subsets of $A$ such that $|\mathcal X|=2^{\kappa}$. For each $U\in \mathcal X$ define $f_U\colon P\to P$ by $f_U(x)=b$ if $x$ is in the filter generated by $U$ in $P$, else $f_U(x)=a$. The set $\{f_U | U\in \mathcal X\}$ is an antichain of size $2^{\kappa}$ in $\textrm{End}(P)$.

For part 4 of the lemma, assume that $W = \{w_{\alpha} : \alpha < \kappa\}$ is an enumeration of an increasing $\kappa$-sequence in $P$. Define a partial function $\sigma\colon P\to P$ by taking $\sigma(x)$ to be the least element $w_{\alpha}$ in $W$ such that $x\not\geq w_{\alpha}$. This $\sigma$ is a monotone total function if $W$ is cofinal in $P$, but only a partial function if there is some $q\in P$ above every $w_{\alpha}$. If such $q$ exists, fix one such, and extend the definition of $W$ to include $w_{\kappa}=q$ and extend the definition of $\sigma$ so that $\sigma(x)=q$ if $x\geq w_{\alpha}$ for all $\alpha<\kappa$. Either way, $W$ is well-ordered, of size $\kappa$, and $\sigma\in \textrm{End}(P)$ maps $P$ into $W$ and $\sigma(P)$ is well-ordered and of size $\kappa$. (Note: $\sigma(P)$ might be a proper subset of $W$.)

For every subset $V\subseteq \sigma(P)$ not containing $w_{\kappa}$ define $g\colon \sigma(P)\to \sigma(P)$ by $g(w_{\alpha})=w_{\alpha}$ if $w_{\alpha}\in \sigma(P)\setminus V$ and $g(w_{\alpha})=w_{\alpha+1}$ if $w_{\alpha}\in V$. If $V_1$ and $V_2$ are incomparable subsets of $\sigma(P)$, then $g_{V_1}$ and $g_{V_2}$ are incomparable elements of $\textrm{End}(P)$. There is a set of $2^{\kappa}$-many pairwise incomparable subsets of the $\kappa$-element set $\sigma(P)\setminus\{w_{\kappa}\}$, so $\textrm{End}(P)$ has an antichain of size at least $2^{\kappa}$. \\

If $P$ is infinite and $P\cong \textrm{End}(P)$, then the size of $P$ is restricted by the following facts.

Lemma. Assume that $P$ is an infinite poset.

1. $P$ has an infinite antichain, an infinite well-ordered subset, or an infinite inversely well-ordered subset.

2. If $|P|>2^{\lambda}$ for some infinite $\lambda$, then $P$ has an antichain, a well-ordered subset, or an inversely well-ordered subset of size $>\lambda$.

3. If $P$ has an antichain of size at least $\kappa$, then $\textrm{End}(P)$ has an antichain of size at least $2^{\kappa}$.

4. If $P$ has a well-ordered or inversely well-ordered subset of size at least $\kappa$, then $\textrm{End}(P)$ has an antichain of size at least $2^{\kappa}$.

In particular, these facts imply that if $P\cong \textrm{End}(P)$, then $|P|\geq \beth_{\omega}$. If also $|P|>2^{\lambda}$ for some infinite $\lambda$, then $|P|>2^{2^{\lambda}}$.

Sketch of proofs.

Part 1 of the Lemma follows from Ramsey's Theorem.

Part 2 of the Lemma follows from the Erdos-Rado Theorem.

For part 3 of the Lemma, if $P$ is an antichain, then $\textrm{End}(P)$ is an antichain of size $2^{|P|}$, so the claim is clear. Otherwise choose $a<b$ in $P$. Let $A\subseteq P$ be an antichain in $P$ of cardinality $\kappa$. It is possible to find a set $\mathcal X$ of pairwise incomparable subsets of $A$ such that $|\mathcal X|=2^{\kappa}$. For each $U\in \mathcal X$ define $f_U\colon P\to P$ by $f_U(x)=b$ if $x$ is in the filter generated by $U$ in $P$, else $f_U(x)=a$. The set $\{f_U | U\in \mathcal X\}$ is an antichain of size $2^{\kappa}$ in $\textrm{End}(P)$.

For part 4 of the lemma, assume that $W = \{w_{\alpha} : \alpha < \kappa\}$ is an enumeration of an increasing $\kappa$-sequence in $P$. Define a partial function $\sigma\colon P\to P$ by taking $\sigma(x)$ to be the least element $w_{\alpha}$ in $W$ such that $x\not\geq w_{\alpha}$. This $\sigma$ is a monotone total function if $W$ is cofinal in $P$, but only a partial function if there is some $q\in P$ above every $w_{\alpha}$. If such $q$ exists, fix one such, and extend the definition of $W$ to include $w_{\kappa}=q$ and extend the definition of $\sigma$ so that $\sigma(x)=q$ if $x\geq w_{\alpha}$ for all $\alpha<\kappa$. Either way, $W$ is well-ordered, of size $\kappa$, and $\sigma\in \textrm{End}(P)$ maps $P$ into $W$ and $\sigma(P)$ is well-ordered and of size $\kappa$. (Note: $\sigma(P)$ might be a proper subset of $W$.)

For every subset $V\subseteq \sigma(P)$ not containing $w_{\kappa}$ define $g\colon \sigma(P)\to \sigma(P)$ by $g(w_{\alpha})=w_{\alpha}$ if $w_{\alpha}\in \sigma(P)\setminus V$ and $g(w_{\alpha})=w_{\alpha+1}$ if $w_{\alpha}\in V$. If $V_1$ and $V_2$ are incomparable subsets of $\sigma(P)$, then $g_{V_1}$ and $g_{V_2}$ are incomparable elements of $\textrm{End}(P)$. There is a set of $2^{\kappa}$-many pairwise incomparable subsets of the $\kappa$-element set $\sigma(P)\setminus\{w_{\kappa}\}$, so $\textrm{End}(P)$ has an antichain of size at least $2^{\kappa}$. \\

We can say something about the sizes of the antichains in infinite posets $P$ satisfying $P\cong \textrm{End}(P)$.

Lemma. If $P$ is an infinite poset, then

1. End($P$) has an infinite antichain, and

2. if $P$ has an infinite antichain of cardinality $\kappa$, then End($P$) has an antichain of cardinality $2^{\kappa}$.

Sketch of proof of item 1. Case 1: $P$ has an infinite antichain. Then End($P$) must also have one, since there is an embedding of $P$ into End($P$) which takes $x$ to the constant function with range $\{x\}$. Case 2: $P$ has no infinite antichain. In this case $P$ must contain an increasing or decreasing $\omega$-sequence. Assume increasing. From this it can be argued that $P$ must have an endomorphism $\sigma\colon P\to P$ whose range $\sigma(P)$ is either an increasing $\omega$-sequence or an increasing $\omega+1$-sequence.

Here is how one produces such a $\sigma$. Suppose that $\Sigma: p_0<p_1<\cdots$ is an increasing $\omega$-sequence. Case 1: $\Sigma$ is cofinal in $P$. In this case, map $x\in P$ to the least $p_i$ such that $x\not\geq p_i$ Case 2: $\Sigma$ is not cofinal, because there is an element $q$ above every $p_i$. In this case, map $x\in P$ to the least $p_i$ such that $x\not\geq p_i$ if such a $p_i$ exists. If $x\geq p_i$ for every $i$, map $x$ to $q$. In Case 1, the range of $\sigma$ is an $\omega$-sequence, while in Case 2 it is an $\omega+1$-sequence.

It is not hard to show that End($\omega$) and End($\omega+1$) each have infinite antichains, so End($\sigma(P)$) has an infinite antichain. If $\{\alpha_0, \alpha_1, \ldots\}$ is an infinite antichain in End($\sigma(P)$), then $\{\alpha_0\circ\sigma, \alpha_1\circ\sigma, \ldots\}$ is an infinite antichain in End($P$).

Sketch of proof of item 2. Case 1: $P$ is an antichain. In this case, End($P$) is an antichain of cardinality $2^{|P|}$, so there is really nothing to do. Case 2: $P$ contains elements $a<b$. Let $A\subseteq P$ be an antichain of cardinality $\kappa$. It is possible to find a set  $\mathcal X$ of pairwise incomparable subsets of $A$ such that $|{\mathcal X}|=2^{\kappa}$. For each $U\in{\mathcal X}$ define $f_U\colon P\to P$ by $f_U(x)=b$ if $x$ is in the filter generated by $U$ in $P$, else $f_U(x)=a$. The set $\{f_U | U\in\mathcal X\}$ is an antichain of size  $2^{\kappa}$ in End($P$). \\

So if $P$ is an infinite poset satisfying $P\cong \textrm{End}(P)$, then it has antichains of size $\beth_n$ for every $n$.

If $P$ is infinite and $P\cong \textrm{End}(P)$, then the size of $P$ is restricted by the following facts.

Lemma. Assume that $P$ is an infinite poset.

1. $P$ has an infinite antichain, an infinite well-ordered subset, or an infinite inversely well-ordered subset.

2. If $|P|>2^{\lambda}$ for some infinite $\lambda$, then $P$ has an antichain, a well-ordered subset, or an inversely well-ordered subset of size $>\lambda$.

3. If $P$ has an antichain of size at least $\kappa$, then $\textrm{End}(P)$ has an antichain of size at least $2^{\kappa}$.

4. If $P$ has a well-ordered or inversely well-ordered subset of size at least $\kappa$, then $\textrm{End}(P)$ has an antichain of size at least $2^{\kappa}$.

In particular, these facts imply that if $P\cong \textrm{End}(P)$, then $P\geq \beth_{\omega}$. If also $|P|>2^{\lambda}$ for some infinite $\lambda$, then $|P|>2^{2^{\lambda}}$.

Sketch of proofs.

Part 1 of the Lemma follows from Ramsey's Theorem.

Part 2 of the Lemma follows from the Erdos-Rado Theorem.

For part 3 of the Lemma, if $P$ is an antichain, then $\textrm{End}(P)$ is an antichain of size $2^{|P|}$, so the claim is clear. Otherwise choose $a<b$ in $P$. Let $A\subseteq P$ be an antichain in $P$ of cardinality $\kappa$. It is possible to find a set  $\mathcal X$ of pairwise incomparable subsets of $A$ such that $|\mathcal X|=2^{\kappa}$. For each $U\in \mathcal X$ define $f_U\colon P\to P$ by $f_U(x)=b$ if $x$ is in the filter generated by $U$ in $P$, else $f_U(x)=a$. The set $\{f_U | U\in \mathcal X\}$ is an antichain of size  $2^{\kappa}$ in $\textrm{End}(P)$.

For part 4 of the lemma, assume that $W = \{w_{\alpha} : \alpha < \kappa\}$ is an enumeration of an increasing $\kappa$-sequence in $P$. Define a partial function $\sigma\colon P\to P$ by taking $\sigma(x)$ to be the least element $w_{\alpha}$ in $W$ such that $x\not\geq w_{\alpha}$. This $\sigma$ is a monotone total function if $W$ is cofinal in $P$, but only a partial function if there is some $q\in P$ above every $w_{\alpha}$. If such $q$ exists, fix one such, and extend the definition of $W$ to include $w_{\kappa}=q$ and extend the definition of $\sigma$ so that $\sigma(x)=q$ if $x\geq w_{\alpha}$ for all $\alpha<\kappa$. Either way, $W$ is well-ordered, of size $\kappa$, and $\sigma\in \textrm{End}(P)$ maps $P$ into $W$ and $\sigma(P)$ is well-ordered and of size $\kappa$. (Note: $\sigma(P)$ might be a proper subset of $W$.)

For every subset $V\subseteq \sigma(P)$ not containing $w_{\kappa}$ define $g\colon \sigma(P)\to \sigma(P)$ by $g(w_{\alpha})=w_{\alpha}$ if $w_{\alpha}\in \sigma(P)\setminus V$ and $g(w_{\alpha})=w_{\alpha+1}$ if $w_{\alpha}\in V$. If $V_1$ and $V_2$ are incomparable subsets of $\sigma(P)$, then $g_{V_1}$ and $g_{V_2}$ are incomparable elements of $\textrm{End}(P)$. There is a set of $2^{\kappa}$-many pairwise incomparable subsets of the $\kappa$-element set $\sigma(P)\setminus\{w_{\kappa}\}$, so $\textrm{End}(P)$ has an antichain of size at least $2^{\kappa}$. \\