The "conservative" class theory, NBG, proves no new theorems about sets (with respect to ZFC). The choice function used here is set choice, and it's not too hard to prove (if M is a ctm for ZFC, then D(M) is a ctm for NBG and has the same set universe).

However, if we add the axiom that there is a bijection (in the class universe) between $\mathbf{V}$ and $\mathbf{ON}$, the classes of sets and ordinals, respectively, this is apparently still conservative over ZFC (with a much stronger class choice axiom). However, I can't find a reference for this.

Apparently this fact is credited to Easton and Solovay, published by Easton in 1964, and apparently it uses class forcing, but I can't find any more specific information on this topic, or the paper itself. Does anyone have more specific information on this, or better search skills than I do?