This is my take on the situation:
The problem is not doing an allopathic residency but not having done an osteopathic rotating internship. For several reasons, which were discussed elsewhere on this board (you may be interested at looking at the past postings), it is very hard to do a rotating internship and then an allopathic residency (the problem is funding). But if you do your residency without having done a rotating internship, then you haven't fulfilled the AOA requirements for your licensure as a DO. No big deal in most states but 5, you just go ahead and apply for your license to the local medical board.
The kicker is really in obtaining licensure in the 5 states where there are 2 different medical licensing boards, one for DOs and one for MDs. Because you didn't do a rotating internship, the DO board doesn't want to license you and because you are a DO and not an MD the allopathic board cannot license you, even though you may have done an allopathic residency. It is a political thing.
As for membership in the AOA, it should not be an overriding concern. Your ability to obtain a license to practice medicine is much more important (isn't that a beautifull understatement?!). I don't know if a person who does an allopathic residency can join the AOA but I don't see why not. This person is still licensed to practice as a DO. In the unlikely event that it is indeed the case, just join the AMA!
Again, if you don't do a rotating internship you cannot practice medicine in 5 states. That is it! Strictly speaking, it has nothing to do with the kind of residency program you do. You could always do your rotating internship and then do an allopathic residency and be licensable in all states, although this is an unlikely scenario.
By the way, the AOA is not going to award hardship exemptions anymore. The decision was just made recently. A couple of recent classes were grandfathered in and can still apply for it, but I am not sure if the class of 2001 can, and I know for sure that my class, 2002, will not be able to. This was done because a decision was made to recognize the 1st year of an allopathic residency as fullfilling the requirement for the rotating internship, AS LONG AS it provides the equivalent training (very unlikely, so it seems a moot point to me).