Yes. I've added a note on that at the end.

The claim $\mu_{\omega}^*(A^{\omega})=1$ is right, and is one of the few things in analysis which seems to require $\text{DC}_{\mathbb{R}}$ and not just $\text{CC}_{\mathbb{R}}.$ (I haven't confirmed $\text{CC}_{\mathbb{R}}$ doesn't imply this principle but it does fail in Feferman-Levy model).

I don’t think you mean to restrict to Lebesgue measurable sets in the smallness definition. Any Lebesgue measurable set of cardinality less than continuum is measure zero.

@AndreasLietz I don't follow. The two different possibilities will have the same parity.

Right that's why I think $Z_2^{\Omega}$ is the "right theory" for choiceless analysis. I expect choiceless analysis of the classes BV, semi-continuous, Baire class to go through under any subsystem of $Z_3$ containing the schema extending $\Pi^1_2$-CA to allow third-order parameters.

That’s not a theorem of $Z_3$ or even ZF. The point of my question is whether choiceless theorems of strong systems tend to remain so over $Z_2^{\Omega}.$