Working in $L$, suppose $L \models \kappa$ is a cardinal and $(\mathbb{P}, <) \in L_\kappa$. Let $\varphi(x)$ be a $\Sigma_1^1$ formula. Let $\tau \in L_\kappa$ be a $\mathbb{P}$-name for an element of ${}^\omega\omega$. Is "$1_\mathbb{P} \Vdash \varphi(\tau)$" absolute between $L_\kappa$ and $L$?

It seems that if $L_\kappa$ satisfies enough of set theory to prove the Mostowski Absoluteness and some basic facts about the forcing relation, then this would be true. For instance if $\kappa$ was inaccessible or even regular uncountable. For any arbitrary cardinal $\kappa$, does $L_\kappa$ satisfy enough set theory for the Mostowski absoluteness.

A related question is what are the known fragments of ZF that all $L_\kappa$ (or the fine structures $J_\kappa$) where $\kappa$ is a cardinal satisfy?

Thanks for any information that can be provided.