If one doesn’t accept the Axiom of Choice but uses instead Dependent Choice the paradox no longer holds. Is it the case in this situation that the 4 desirable properties hold?
The Banach-Tarski paradox shows that classical set theory makes the wrong assumptions to intrinsically model measure theory and probability.
There are other systems which don't suffer from this paradox and hence don't need all the machinery of sigma algebras and measurable sets.
I wish there was a good accessible book/article/blog post about this, but as is you'd have to Google point-free topology or topos of probability (there are several).
Is there a known set theory of the form ZF+(something) which relatively consistent with ZFC in which additive, isometry invariant measures exist?
I guess what you are saying is that the only known, reasonable way around this is the topos notion you mentioned.