But isn't Banach-Tarski one of the main reasons why AoC is highly contested in the first place? (I don't know much about higher math, so this only my amateur impression.)

It's also contested because while intuitive to a countable set of countable sets, generalizing to higher cardinals is not intuitive. Although the same could be said of the Power Law as well...

The paradoxical decomposition of the free group doesn’t rely on choice though, and that’s arguably the “heart” of BT.