lots of mathematicians reject AC

No they don't (*). Your saying this explicitly somewhat confirms my brain's natural, automatic assumption that your error here (and in similar comments in the past -- "infinite set atheism" and all that business) is as much sociological as philosophical: all along, I instinctively thought, "he doesn't seem to realize that that's a low-status position".

ZFC is considered the standard axiom system of modern mathematics. I have no doubt that if an international body (say, the IMU) were to take a vote and choose a set of "official rules of mathematics", the way (say) FIDE decides on the official rules of chess, they would pick ZFC (or something equivalent).

Now it's true, there are some mathematicians who are contrarians and think that AC is somehow "wrong". They are philosophically confused, of course; but, more to the point here in this comment, they are a marginal group. (In fact, even worrying about foundational issues too much -- whatever your "position" -- is kind of a low-status marker itself: the sociological reality of the mathematical profession is that members are expected to get on with the business of proving impressive-looking new theorems in mainstream, high-status fields, and not to spend time fussing about foundations except at dinner parties.)

See also this comment of mine.

(*) I don't know the numbers, or how you define "lots", and there are a large number of mathematicians in the world, so technically I don't know if it's literally false that "lots" of mathematicians would say that they "reject AC" . But the clear implication of the statement -- that constructivism is a mainstream stance -- most definitely is false.