Well, after some more research I'm fairly certain the "fish" I saw swimming around were scuds. Basically looked like the tiniest shrimp ever. I read that bettas will eat them, but I also read they multiply like crazy. Also, they're liable to carry diseases and parasites that can infect larger animals (e.g. fish).
I ended up doing a hydrogen peroxide dip, and that seemed to get them all. All in all I'd say there were about 30 in there! I'm the kind of person who can often get insanely frustrated. By that point I didn't care about snails anymore. I rinsed the moss thoroughly with conditioned water (at least 2x the recommended amount) about six times. May have been overkill, but better safe than sorry. I didn't see anymore scuds after the dip.
It's in my tank now, so we'll see how it goes. I really liked this LFS. All their betts are kept in large tanks. I think the smallest tank I saw a betta in was probably a 1 gallon. Their husbandry is excellent for all their animals. I just wish someone would have told me that snails weren't my only worry with hitchikers on the moss. Not sure if this experience is typical, but this was definitely not the case at another LFS in my area.
Anyway, moss in place. If I end up with snails, I'll deal with that then! Thanks for the advice everyone!