It's well-known that if a cardinal is weakly compact, then it is weakly compact in $L$. Seems natural to ask if weak-compactness is preserved for arbitrary inner models. Since I've never heard this, I'm guessing the answer is no, but I'm having trouble finding a counterexample.
I have something that I think might work, but for the ZF version of the problem:
Since it's assigned as an exercise in Jech's AC book I believe that "if $\kappa$ is weakly compact, then it's weakly compact in L" still holds in ZF. So let $\omega_1$ be weakly compact (which is consistent with ZF). Then, go to an $L[A]$ that correctly computes $\omega_1.$ We'll have AC in $L[A]$, thus $\omega_1$ can't be weakly compact in $L[A].$
(But is it consistent with $\omega_1$ being weakly compact that there's an $L[A]$ that correctly computes $\omega_1$? I think the latter requires some choice on the reals.)
If this works, then maybe we can get something in ZFC from the Jech model (i.e. start with $\kappa$ weakly compact, collapse $\kappa$ to be countable and symmetrize so that $\kappa$ is $\omega_1$ in the symmetric submodel)... perhaps we can make some partial extension where $\kappa$ isn't fully collapsed and is still weakly compact?
My question is mostly just the title. Would also appreciate feedback on my attempt. (But, even if viable, it seems convoluted, so would guess there's a better example.)
The problem with your argument is that if $\omega_1$ is weakly compact, then there is no $A$ such that $L[A]$ computes $\omega_1$ correctly. To see this, note that if $M$ is an inner model of $\sf ZFC$ which computes $\omega_1$ correctly, then $M$ knows a special Aronszajn tree (specifically the canonical construction of an Aronszajn tree using the rationals), but since in $V$ there are no such trees, the tree must have a branch, which must have collapsed $\omega_1$.
This is not too dissimilar from the case where $\omega_1$ is singular. Given any inner model of $\sf ZFC$, it cannot possibly compute $\omega_1$ correctly.
To your general question, it is not even necessarily the case that weak compactness is preserved into inner models of $\sf ZFC$. Case in point, start with $L$ where $\kappa$ is weakly compact, then using an Easton support iteration, add a Cohen subset to every regular cardinal below $\kappa$. This, in effect, adds a $\kappa$-Aronszajn tree, and so it kills the weak compactness of $\kappa$. However, if we add a Cohen subset to $\kappa$, effectively adding a branch through this tree, we resurrect the weak compactness of $\kappa$.
And so, we have $L\subseteq L[G_0]\subseteq L[G_0][G_1]$, where $\kappa$ is weakly compact in $L$ and in $L[G_0][G_1]$, but not in $L[G_0]$.