Perhaps ironically, despite appearances, the process you propose does not depend on the axiom of choice.
This is because we can prove, in the small and generally trusted metatheory PRA, that ZFC is inconsistent if and only if ZF (= ZFC − AC) is inconsistent (if and only if IZF (= ZFC − AC − LEM) is inconsistent).
[ This metaproof rests on the fact that ZF can prove that the axiom of choice (AC) holds in "Gödel's sandbox" L, the "constructible universe", even if it might not hold in the universe of all sets. ]
In other words: Adding the axiom of choice to ZF doesn't cause new inconsistencies. In case ZF is consistent (a statement which most logicians believe), then ZFC is so as well.
This is because we can prove, in the small and generally trusted metatheory PRA, that ZFC is inconsistent if and only if ZF (= ZFC − AC) is inconsistent (if and only if IZF (= ZFC − AC − LEM) is inconsistent).
[ This metaproof rests on the fact that ZF can prove that the axiom of choice (AC) holds in "Gödel's sandbox" L, the "constructible universe", even if it might not hold in the universe of all sets. ]
In other words: Adding the axiom of choice to ZF doesn't cause new inconsistencies. In case ZF is consistent (a statement which most logicians believe), then ZFC is so as well.
A couple pointers to the literature are here: https://www.speicherleck.de/iblech/stuff/37c3-axiom-of-choic...