BNGetToonInfo and
BNGetFriendToonInfo both provide a realmID, though as I only have access to a trial account at the moment, I can't actually check whether it's different for the US and EU realms of the same name. If it is, then you could set up a hardcoded table mapping each realmID to its region, and then check the player's Battle.net friends (assuming they have any) to find out which region they were playing on.
If the realmID proves useful, it might also be worth asking Blizzard to add it to non-Battle.net functions that currently return a realm name, eg. UnitName, UnitFullName, GetRealmName, though I'm not sure how much they really listen to that kind of suggestion...