The appropriate questions are: who is promoting this story and why? It may be the real deal (knowing what you already know about Michelle Obama, is it really that much of a stretch?) ... or it might be disinformation designed to discredit conservative news sources that prematurely promote it. We'll see.
Obama has enemies. Some of them freelance, being sympathetic with his political opponents, but not part of any organization. The most prevalent speculation, compatible with "the event is fiction," is that Hillary! supporters are running a whisper campaign. Saying the tape was found by and is being held by Republicans is a way to play "clean hands."
And as you note, the rumor is not "beyond all credibility." All "good rumors" (meaning the false ones that draw suckers) have a whiff of truth about them.