It should be emphasized that this theorem is quite old and in fact reasonably well understood. We have quite closely identified how this "paradox" arises. It can be arguably seen as some kind of ghost or triviality created by our subtle choice of axioms. If we switch AOC for something similiar like axiom of dependent choice, the "paradox" vanishes, because we lose the ability to even access these niche sets. The proof of Banach-Tarski itself is a bit involved for me at least, but there are similar results which are less fancy but use much simpler reasoning that illustrate the core of the problem. I don't know much about your competency in proof-reading, but if you can half-follow that proof, I'd recommend the "Example" on this page:
In summary, AOC is used to divide a circle into countably infinitely many identical pieces. Summing these pieces up like a series gives us a problem - if the length of each piece is 0 then the sum, the length of the circle, is 0. But if each piece has a constant positive length, the circle's length becomes infinite. The description in the example uses complex numbers and group theory (unnecessarily, tbh) which makes it seem a lot more esoteric than it is. In reality, it is something that can be understood without much background - assuming one understands AOC. If you're interested, I can try writing a more fleshed out description which assumes less background.
Maybe I'm speaking more assertively than necessary - you certainly could resolve this "paradox" in a different way. I guess you could alter the definition of volume so that it does not demand volume is invariant under rotations / partitions / etc. Or maybe you could even keep volume the same and alter arithmetic itself, though I honestly can't even imagine such a thing. Nonetheless, from what I know, ppl typically just don't bother attempting to assign these sets a volume, in the same way we typically don't bother trying to assign a literal value to 1/0. Running away from our problems or pretending they don't exist is an unreasonably effective strategy lol.
I shouldn't really say much about the science because I'm sure you know a lot more about that than me. Nonetheless, I at least agree with the sentiment that we should try to be humble and alert for errors in our thinking and that common sense assumptions about reality, even if they seem reasonable initially, may run contrary to future observations.