So, this is actually impossible to do.
Since exceptions are c-level classes, adding members to ValueError means anything that multiple-inherits from ValueError and any other complex exception gets a multiple bases layout conflict. There's already one such class - _pyio.UnsupportedOperation inherits both ValueError and OSError - but the bigger issue is that this would break compatibility.

After discussing it (face-to-face) with a bunch of core devs, I think the only reasonable course of action is to close the issue and never speak of this again.