Solovay's model proves that an inaccessible cardinal exists

By F1, every $\alpha <\kappa$ has in $V[G]$ a real coding a surjection $\omega \twoheadrightarrow \alpha$; F2 puts each code in both $M$ and $L(\mathbb{R})$, so every such $\alpha$ is internally countable. Conversely, an internal surjection $\omega \twoheadrightarrow \kappa$ would belong to $V[G]$, contradicting F1 there. Thus the designated construction ordinal is ${\omega }_1$ in both inner models and is not inaccessible.

F3 has logical form $\mathrm{Con}(T_{\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{c}\mathrm{c}})\to \mathrm{Con}(T_{\mathrm{r}\mathrm{e}\mathrm{g}})$. It neither reverses this arrow nor inserts “there is an inaccessible” into $T_{\mathrm{r}\mathrm{e}\mathrm{g}}$. The construction also does not prove that no other ordinal can be inaccessible in a chosen target model; that stronger assertion is not needed.

Hence both the proposed survival of the designated $\kappa$ and the inference from relative consistency to an internal inaccessible are invalid.