I thought ZFC proved the existence of an inductive well-ordering that is itself a set for any stage of V. NBG with only the regular AC should then prove/assert the existence of a class R of ordered pairs (a,b) such that either:
a has lesser rank than b, or:
they both have rank α and (a,b) exists in the ZFC well-ordering of the complement of V(α) within V(α+1).
I don't see that we have to quantify over classes there. By another phrasing it just unites for all V(β) their inductive well-orderings along these lines, which are already supersets of the same ordering for any previous stage. And then NBG should easily prove this well-orders V. How does this proof fail?