“An interesting question is what to do if a Chabad woman goes to a place where they pasken that wigs are halachicly prohibited (and no woman there wears only a wig in public.)”
There are actually such places, and this would be an interesting question if it weren’t about Chabad. It is very well known that Lubavitchers are never choshesh for other’s minhagim no matter the circumstance. I do not know their heter, but it’s clearly their established way.