Work over $\mathsf{GB}$, Gödel-Bernays set theory (without choice). My question is the following:

Question. Is the following statement consistent with $\mathsf{GB}$? There is a universal class for elementary embeddings $J\subseteq V\times V$ such that if $j\colon V\to V$ is ($\Sigma_1$-)elementary, then there is a set $a\in \operatorname{dom} J$ such that $j=J_a$, where $J_a=\{b \mid (a,b)\in X\}$.

Note that the above principle can consistently fail if there is a club Berkeley cardinal:

Proposition. Suppose that there is a club Berkeley cardinal. Then $\delta$ is a totally Reinhardt cardinal in a strong sense that $(V_\delta,V_{\delta+1})$ is a model of the second-order $\mathsf{ZF}$ and that for every $A\subseteq V_\delta$ and $\alpha<\delta$ there is an $A$-elementary embedding $j\colon (V_\delta,A)\to (V_\delta,A)$ such that $j^+[A]=A$ and $\operatorname{crit}(j)>\alpha$.

(This follows from a proof of Theorem 3.8 of Large Cardinals Beyond Choice by Bagaria-Koellner-Woodin.)

Proposition. Working over $\mathsf{GB}$, assume that for each class $A$ there is a ($\Sigma_1(A)$-) elementaty embedding $j\colon (V,A)\to (V,A)$ and $A=j^+[A]$. Then there is no universal class for elementary embeddings.

Proof. Suppose that there is such $J$ and let $j\colon (V,J)\to (V,J)$. Then $j=J_p$ for some $p$. Say $q$ is bad if $J_q$ codes a ($\Sigma_1(J)$-)elementary embedding $j\colon (V,J)\to (V,J)$.

Now we can carry over Suzuki's proof for the non-existence of definable elementary embeddings, so we have a contradiction: Let $\kappa_0$ be the least $\kappa$ such that there is bad $q$ such that $\kappa$ is a critical point of $J_q$. On the one hand, $\kappa_0$ is a critical point of $J_{q_0}$ for some $q_0$, so $J_{q_0}(\kappa_0)>\kappa_0$. On the other hand, $\kappa_0$ is definable by a formula with $J$, so it must be preserved by $J_{q_0}$, a contradiction.

The motivation of my question is what are the class of definable elementary embeddings from a given elementary embedding $j\colon V\to V$ and set parameters. The only practical way I know for obtaining new elementary embedding from the given ones is using composition and $j,k\mapsto j^+[k]$. It brings me to the following question:

Question. Let $I$ be the set of names for the free algebra generated by $\mathsf{j}$: that is, $I$ is the least set of finite strings satisfying the following rules:

1. $\mathsf{j}\in I$,
2. If $x,y\in I$, then $x\circ y,\ x\cdot y\in I$.

Is it consistent with $\mathsf{GB}$ with the existence with the class $J\subseteq I\times V$ satisfying the following conditions?

1. $J_x$ is a ($\Sigma_1$-)elementary emebdding from $V$ to $V$ for every $x\in I$.
2. $J_{x\circ y}=J_x\circ J_y$, and
3. $J_{x\cdot y} = J_x^+[J_y]$.
4. For every ($\Sigma_1$-)elementary embedding $j\colon V\to V$, there is $x\in I$ such that $j=J_x$.

(Intuitively, the above statement claims all elementary $j\colon V\to V$ is a member of a free algebra generated from $J_{\mathsf{j}}$.)