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It suffices to show this for l.o. free monoids on finite alphabets, as follows from the Compactness Theorem in logic - cf. any text on First Order Logic. [This principle can be applied to a range of similar problems.]

Indeed, let $\mathbb M = (M, \cdot, \le)$ be a linearly ordered free monoid on the alphabet $X$, and consider the first-order language $\mathcal{L} = (\circ, \preceq, \{x_m|m \in M\})$ consisting of a binary function symbol $\circ$ to represent multiplication, a binary relation symbol $\preceq$ representing ordering, plus an individual constant symbol $x_m$ for each element $m$ of the set $M$.

Then let $T$ be the $\mathcal{L}$-theory having the following axioms:

1. the usual axioms for linearly ordered groups, expressed using $\circ$ and $\preceq$
2. $x_m \ne x_n$ for all $m, n\in M$ with $m\ne n$ [these axioms ensure that $M$ naturally injects as a subset of any model of $T$ via the interpretation of the $x_m$ constants]
3. $x_m \preceq x_n$ for all $m, n\in M$ with $m\le n$ [these make this injection an embedding of l.o. sets]
4. $x_m \circ x_n = x_{m\cdot n}$ for all $m, n\in M$ [which make the embedding a monoid homomorphism].

So any model $\mathbb G = (G, \cdot, \le, \{g_m|m \in M\})$ of $T$ will provide a desired l.o. group, with $m\mapsto g_m$ (being the interpretation of the constant $x_m$ in $\mathbb G$) giving the desired embedding $\mathbb M\to \mathbb G$. And conversely.

By the Compactness Theorem, the theory $T$ admits a model iff every finite subset $\Delta$ of the axioms of $T$ does. Now only finitely many symbols $x_m$ can occur in sentences belonging to such a $\Delta$, and the finitely many elements $m\in M$ so involved can be expressed as words in a finite subalphabet $Y\subseteq X$. This $Y$ generates a l.o. free submonoid $\mathbb S$ of $\mathbb M$ that contains all the $m\in M$ for which $x_m$ figures in statements occurring in $\Delta$. And so any l.o. group that "extends" $\mathbb S$ (in the sense of the OP) will be a model of $\Delta$.

Hence it is enough to consider free l.o. monoids on finite alphabets.

It suffices to show this for l.o. free monoids on finite alphabets, as follows from the Compactness Theorem in logic - which can be found in any text on First Order Logic. [This principle can be applied to a range of similar problems.]

Indeed, let $\mathbb M = (M, \cdot, \le)$ be a linearly ordered free monoid on the alphabet $X$, and consider the first-order language $\mathcal{L} = (\circ, \preceq, \{x_m|m \in M\})$ consisting of a binary function symbol $\circ$ to represent multiplication, a binary relation symbol $\preceq$ representing ordering, plus an individual constant symbol $x_m$ for each element $m$ of the set $M$.

Then let $T$ be the $\mathcal{L}$-theory having the following axioms:

1. the usual axioms for linearly ordered groups, expressed using $\circ$ and $\preceq$
2. $x_m \ne x_n$ for all $m, n\in M$ with $m\ne n$ [these axioms ensure that $M$ naturally injects as a subset of any model of $T$ via the interpretation of the $x_m$ constants]
3. $x_m \preceq x_n$ for all $m, n\in M$ with $m\le n$ [these make this injection an embedding of l.o. sets]
4. $x_m \circ x_n = x_{m\cdot n}$ for all $m, n\in M$ [which make the embedding a monoid homomorphism].

So any model $\mathbb G = (G, \cdot, \le, \{g_m|m \in M\})$ of $T$ will provide a desired l.o. group, with $m\mapsto g_m$ (being the interpretation of the constant $x_m$ in $\mathbb G$) giving the desired embedding $\mathbb M\to \mathbb G$. And conversely.

By the Compactness Theorem, the theory $T$ admits a model iff every finite subset $\Delta$ of the axioms of $T$ does. Now only finitely many symbols $x_m$ can occur in sentences belonging to such a $\Delta$, and the finitely many elements $m\in M$ so involved can be expressed as words in a finite subalphabet $Y\subseteq X$. This $Y$ generates a l.o. free submonoid $\mathbb S$ of $\mathbb M$ that contains all the $m\in M$ for which $x_m$ figures in statements occurring in $\Delta$. And so any l.o. group that "extends" $\mathbb S$ (in the sense of the OP) will be a model of $\Delta$.

Hence it is enough to consider free l.o. monoids on finite alphabets.

[Incidentally, by the same token, if all finitely generated free groups are linearly orderable, so are all free groups.]

It suffices to show this for l.o. free monoids on finite alphabets, as follows from the Compactness Theorem in logic - cf. any text on First Order Logic. [This principle can be applied to a range of similar problems.]

Indeed, let $\mathbb M = (M, \cdot, \le)$ be a linearly ordered free monoid on the alphabet $X$, and consider the first-order language $\mathcal{L} = (\circ, \preceq, \{x_m|m \in M\})$ consisting of a binary function symbol $\circ$ to represent multiplication, a binary relation symbol $\preceq$ representing ordering, plus an individual constant symbol $x_m$ for each element $m$ of the set $M$.

Then let $T$ be the $\mathcal{L}$-theory having the following axioms:

1. the usual axioms for linearly ordered groups, expressed using $\circ$ and $\preceq$
2. $x_m \ne x_n$ for all $m, n\in M$ with $m\ne n$ [these axioms ensure that $M$ naturally injects as a subset of any model of $T$ via the interpretation of the $x_m$ constants]
3. $x_m \preceq x_n$ for all $m, n\in M$ with $m\le n$ [these make this injection an embedding of l.o. sets]
4. $x_m \circ x_n = x_{m\cdot n}$ for all $m, n\in M$ [which make the embedding a monoid homomorphism].

So any model $\mathbb G = (G, \cdot, \le, \{g_m|m \in M\})$ of $T$ will provide the desired l.o. group, with $m\mapsto g_m$ (the interpretation of the constant $x_m$ in $\mathbb G$) giving the desired embedding $\mathbb M\to \mathbb G$. And conversely.

By the Compactness Theorem, the theory $T$ admits a model iff every finite subset $\Delta$ of the axioms of $T$ does. Now only finitely many symbols $x_m$ can occur in sentences belonging to such a $\Delta$, and the finitely many elements $m\in M$ so involved can be expressed as words in a finite subalphabet $Y\subseteq X$. This $Y$ generates a l.o. free submonoid $\mathbb S$ of $\mathbb M$ that contains all the $m\in M$ involved for which $x_m$ figures in statements occurring in $\Delta$. And so any l.o. group that "extends" $\mathbb S$ (in the sense of the OP) will be a model of $\Delta$.

Hence it is enough to consider free l.o. monoids on finite alphabets.

It suffices to show this for l.o. free monoids on finite alphabets, as follows from the Compactness Theorem in logic - cf. any text on First Order Logic. [This principle can be applied to a range of similar problems.]

Indeed, let $\mathbb M = (M, \cdot, \le)$ be a linearly ordered free monoid on the alphabet $X$, and consider the first-order language $\mathcal{L} = (\circ, \preceq, \{x_m|m \in M\})$ consisting of a binary function symbol $\circ$ to represent multiplication, a binary relation symbol $\preceq$ representing ordering, plus an individual constant symbol $x_m$ for each element $m$ of the set $M$.

Then let $T$ be the $\mathcal{L}$-theory having the following axioms:

1. the usual axioms for linearly ordered groups, expressed using $\circ$ and $\preceq$
2. $x_m \ne x_n$ for all $m, n\in M$ with $m\ne n$ [these axioms ensure that $M$ naturally injects as a subset of any model of $T$ via the interpretation of the $x_m$ constants]
3. $x_m \preceq x_n$ for all $m, n\in M$ with $m\le n$ [these make this injection an embedding of l.o. sets]
4. $x_m \circ x_n = x_{m\cdot n}$ for all $m, n\in M$ [which make the embedding a monoid homomorphism].

So any model $\mathbb G = (G, \cdot, \le, \{g_m|m \in M\})$ of $T$ will provide a desired l.o. group, with $m\mapsto g_m$ (being the interpretation of the constant $x_m$ in $\mathbb G$) giving the desired embedding $\mathbb M\to \mathbb G$. And conversely.

By the Compactness Theorem, the theory $T$ admits a model iff every finite subset $\Delta$ of the axioms of $T$ does. Now only finitely many symbols $x_m$ can occur in sentences belonging to such a $\Delta$, and the finitely many elements $m\in M$ so involved can be expressed as words in a finite subalphabet $Y\subseteq X$. This $Y$ generates a l.o. free submonoid $\mathbb S$ of $\mathbb M$ that contains all the $m\in M$ for which $x_m$ figures in statements occurring in $\Delta$. And so any l.o. group that "extends" $\mathbb S$ (in the sense of the OP) will be a model of $\Delta$.

Hence it is enough to consider free l.o. monoids on finite alphabets.

It suffices to show this for l.o. free monoids on finite alphabets, as follows from the Compactness Theorem in logic - cf. any text on First Order Logic. [This principle can be applied to a range of similar problems.]

Indeed, let $\mathbb M = (M, \cdot, \le)$ be any linearly ordered free monoid, and consider the first-order language $\mathcal{L} = (\circ, \preceq, \{x_m|m \in M\})$ consisting of a binary function symbol $\circ$ to represent multiplication, a binary relation symbol $\preceq$ representing ordering, plus an individual constant symbol $x_m$ for each element $m$ of the set $M$.

Then let $T$ be the $\mathcal{L}$-theory having the following axioms:

1. the usual axioms for linearly ordered groups, expressed using $\circ$ and $\preceq$
2. $x_m \ne x_n$ for all $m, n\in M$ with $m\ne n$ [these axioms ensure that $M$ naturally injects as a subset of any model of $T$ via the interpretation of the $x_m$ constants]
3. $x_m \preceq x_n$ for all $m, n\in M$ with $m\le n$ [these make this injection an embedding of l.o. sets]
4. $x_m \circ x_n = x_{m\cdot n}$ for all $m, n\in M$ [which make the embedding a monoid homomorphism].

So any model $\mathbb G = (G, \cdot, \le, \{g_m|m \in M\})$ of $T$ will provide the desired l.o. group, with $m\mapsto g_m$ (the interpretation of the constant $x_m$ in $\mathbb G$) giving the desired embedding $\mathbb M\to \mathbb G$. And conversely.

By the Compactness Theorem, the theory $T$ admits a model iff every finite subset $\Delta$ of (the axioms of) $T$ does. Now only finitely many symbols $x_m$ can occur in sentences belonging to such a $\Delta$, and the finitely many elements $m$ involved generate a l.o. submonoid $\mathbb S$ of $\mathbb M$ which is necessarily a free monoid on a finite alphabet. Hence it is enough to consider finitely generated free l.o. monoids.

It suffices to show this for l.o. free monoids on finite alphabets, as follows from the Compactness Theorem in logic - cf. any text on First Order Logic. [This principle can be applied to a range of similar problems.]

Indeed, let $\mathbb M = (M, \cdot, \le)$ be a linearly ordered free monoid on the alphabet $X$, and consider the first-order language $\mathcal{L} = (\circ, \preceq, \{x_m|m \in M\})$ consisting of a binary function symbol $\circ$ to represent multiplication, a binary relation symbol $\preceq$ representing ordering, plus an individual constant symbol $x_m$ for each element $m$ of the set $M$.

Then let $T$ be the $\mathcal{L}$-theory having the following axioms:

1. the usual axioms for linearly ordered groups, expressed using $\circ$ and $\preceq$
2. $x_m \ne x_n$ for all $m, n\in M$ with $m\ne n$ [these axioms ensure that $M$ naturally injects as a subset of any model of $T$ via the interpretation of the $x_m$ constants]
3. $x_m \preceq x_n$ for all $m, n\in M$ with $m\le n$ [these make this injection an embedding of l.o. sets]
4. $x_m \circ x_n = x_{m\cdot n}$ for all $m, n\in M$ [which make the embedding a monoid homomorphism].

So any model $\mathbb G = (G, \cdot, \le, \{g_m|m \in M\})$ of $T$ will provide the desired l.o. group, with $m\mapsto g_m$ (the interpretation of the constant $x_m$ in $\mathbb G$) giving the desired embedding $\mathbb M\to \mathbb G$. And conversely.

By the Compactness Theorem, the theory $T$ admits a model iff every finite subset $\Delta$ of the axioms of $T$ does. Now only finitely many symbols $x_m$ can occur in sentences belonging to such a $\Delta$, and the finitely many elements $m\in M$ so involved can be expressed as words in a finite subalphabet $Y\subseteq X$. This $Y$ generates a l.o. free submonoid $\mathbb S$ of $\mathbb M$ that contains all the $m\in M$ involved for which $x_m$ figures in statements occurring in $\Delta$. And so any l.o. group that "extends" $\mathbb S$ (in the sense of the OP) will be a model of $\Delta$. 

Hence it is enough to consider free l.o. monoids on finite alphabets.