Let $V$ be a model of $ZFC$. Let $M \in V$ be (a possibly not transitive) countable model of large fragments of $ZFC$ (for example countable substructures of large $H_\Theta$). If $M$ models enough of $ZFC$, is it possible that $\text{Th}_{\in}(M) \notin M[G]$ for all $G$ generic over $M$ for some fixed forcing poset?
Here $\text{Th}_\in(M)$ just denotes all the $\{\in\}$ sentences true in $M$.
The motivation for this question is: All countable $\in$-structures on $\omega$ can be coded by reals. Fix such a coding. Is it possible to find a countable elementary substructure of a large structure such that $M[G]$ can not contain the code for a model isomorphic to $M$.
Containing the code for $M$, is like saying $M \in M[G]$ which is certainly not possible. However, $M[G]$ does not have the isomorphism that witnesses that this structure on $\omega$ is isomorphic to $M$. However, having a structure isomorphic to $M$ does imply that $\text{Th}_\in(M) \in M[G]$. So this question can be resolved if there is an $M$ such that $Th_\in(M) \notin M[G]$.
Thanks for any help.