Even if it's not discrete at the bottom of QFT or something, it is discrete at the level where physics is applied in classical mechanics.
Conversely, classical mechanics is discrete at the level where physics is applied. That it is applied to realia goes without saying. But it is not applied to reality as a whole.
But as a counter to the assertion that "AC must be wrong because we can't really experience Banach-Tarski in the real world", it's perfectly valid to comment that "well, we also can't really observe continuous functions R -> R either". Or have you ever seen a least upper bound in real life?