Joseph,
“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.)”
Whilst there is no such place nowadays, but in theory the Magen G’Borim paskens clearly that you need to follow that halacha, as it is incl in ‘Das Yehudis’ (see MB in Hilchos Krias Shma ch 74)