If there were a single, valid, BBS ID per Game ID, then, yes, this would be doable. You might not, necessarily, want to do it, but you could.
As the BBS is open to everybody who registers in the BBS, not necessarily a subscriber to AH, then it would be impractical to try to associate which game-ids match the bbs-ids.
Yes, there are other games that tie your BBS/Forum ID to your user ID, AH isn't one of those.
Having the BBS restricted to game members only has been rather frowned upon, in the past.. and I know the reputations of games that did that were considered 'damaged' by many.