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Because measurable cardinals are limits of inaccessible cardinals and, if $\lambda<\lambda'<\kappa$ is inaccessible in $M$, then $\lambda'$ is inaccessible in $(V_\kappa)^M$.
@n901 Yeah, that's a good question. I don't have a metatheorem to list. The intuition I can give in broad strokes is that many of these Borel nonreducibility results are not obtained by, say, an argument that verifies by induction on $\alpha$ that if a function is $\mathbf\Sigma^0_\alpha$, then it is not a reduction from $E$ to $F$. Instead, for instance, Baire category arguments are used, which are more amenable to a forcing approach. Some of the results in the paper by Andretta, Hjorth and Neeman on "effective cardinals" use sophisticated arguments in this spirit.
@user534667 Yeah, Richard and I did a little bit of work on $S_1$ and some other cardinals from Hugh's paper, but it was all very preliminary, I think there is a lot more to explore in this area. William Chan and Steve Jackson recently published some work here. But there is much we don't know. I don't think we even know a reasonable bound on the number of (immediate) successors of $|\mathbb R|$.
(Also, Itay Neeman and Zach Norwood have some nice "triangular embedding theorems" that should allow one to lift the Borel cardinality results via a different route than my approach with Richard.)
(Woodin's argument are more combinatorial and native to the $\mathsf{AD}_{\mathbb R}$ setting, so I am not sure whether they can be adapted to just $\mathsf{AD}^+$, even in natural models.)
Borel cardinality results of the sort discussed here transfer to the $\mathsf{AD}^+$ setting, and we have a good understanding of how to verify claims of this kind. Richard and I wrote a paper a lifetime ago illustrating this with Glimm–Effros-type dichotomies. We never published it, but see our published trichotomy paper. The Borel version of such a result holds in small $\mathsf{ZFC}$ models within the determinacy model, and an ultraproduct argument and Vopěnka forcing allows us to lift it.
