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I get your point, but this is a bad example - the reason (AC) is an axiom and not a theorem is because you can reject it and still have a self-consistent system of mathematics at the end. There's stuff you can't prove without it, but there's much more you can still do without it, and also proving exactly which theorems do or don't depend on it is sometimes an interesting contribution to mathematics in itself. That's probably a politically acceptable category of contribution, too, despite flirting with (AC), if the results are phrased like "Look what foolish falsehoods the imperialist oppressors could have tricked you into if not for Dear Leader's wisdom that Choice Is A Lie!"
Rejecting an actual theorem in this fashion would be a disaster. The Principle of Explosion is bad enough that I wouldn't be surprised to see mathematicians deciding that they'd rather go out in full Kaczynski style with non-metaphorical explosions.
I think the important thing for avoiding political targeting of a field's theories isn't whether the field is "pure", but whether its complex-to-apply theories lead to simple-to-verify results. Math often does this, with problems whose solutions require advanced math to find but only basic math to check, but really much-less-pure engineering is the king here. Eventually either a nuke goes boom or it doesn't; either a rocket goes up or it doesn't. Being good at engineering rockets eventually got Korolev a reprieve from the Gulag, whereas being a perfect historian or economist would have probably made his fate worse.
I mean your comment kind of acknowledges that using or not using AC is something that most mathematicians do mostly as a thought experiment, and that they don't have a personal, ideological bias for or against it. In other words, rejecting all proofs that use AC on purely ideological grounds (dear-leader-style) is flirting with quackery.
Agreed, up to the point that "simple-to-verify" can also lead to ideological ambiguity, e.g. is the Earth's oblate spheroid shape or even gravity "simple-to-verify"?
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