I flipped down the passenger visor on to hold up a sun shade and the whole mirror assembly fell out of the visor. One of the tabs that holds it in appears to be broken. Is it at all possible just to purchase the mirror and cover part or am I going to have to replace the entire visor?

If you were still under warranty it would be covered, if not you need to replace the whole visor for about $85 at the dealer, maybe less online. Same thing happened to me but warranty covered it. It's a pretty common problem, cheap design...

I'd try to go to a junk yard or find a totaled MINI on ebay. You could pull the part yourself for probably under $20. That's what I would do and it's what I've done for past cars.

Mine also broke but I keep forgetting to tell the guys about it when I get the car serviced to get it replaced.

EDIT:
Also, according to RealOEM the visor and mirror pieces are all one part number, that seems kinda strange. So you'll have to order the whole thing, not just the mirror.

Just went through this. Driver's side mirror cover wouldn't stay closed, so in use, the light turns on. Got replacement (carbon black/anthracite) for about $100 (stupid expensive) and the new one comes, super easy to install, but the damn "warning" stickers changed in the replacement part. So now I have a passenger and driver side with different stickers. So I removed both labels. Easy to do with rubbing alcohol and a little patience. Highly suggest removing them from the car if you do this!