You probably could, though you would probably have to give up or weaken other axioms as well. You would need to prevent there being any bijections between the

You probably could, though you would probably have to give up or weaken other axioms as well. You would need to prevent there being any bijections between the good sets and the bad sets, because you then you could use the bijection to define a choice function on the bad set. What set theorists usually do is just distinguish between when you use the axiom of choice and when you don't. There's one super-subtle point, though: even if you ban using the axiom of choice, you can still construct sets where it is independent of the axioms of set theory whether they are measurable. You have to add "these sets are measurable" as an axiom. The simplest way to do it is to consider a class of sets where you know you didn't use the axiom of choice, and then declare that these sets are measurable. One common choice is the class of "projective sets". Usually set theorists impose a stronger property called determinacy, so they add to set theory the Axiom of Projective Determinacy. Then you can say "If I stick to the good sets -- projective sets -- nothing bad happens. Arbitrary choice functions take me out of the projective sets, where bad things can happen."