Sorry to hear about that man...
I have some experience in this, but from the other side and it's usually something really concrete (like you said you had a degree and the background check came back that you did not) but even so we give the candidate a chance to explain and work with them. We had a young guy last year just graduated and we hired him. His BC came back and it said he hadn't graduated so we asked him and it turned out his college required internship credit before he could graduate. So we turned him into an intern until he satisfied the requirement, get his degree, then turned him to regular status.
If it's a reference thing, then it sounds like this firm is hardcore and not even willing to work with you on it.
Good luck man.