As long as you supplement both the fish's and the snail's diet with vegetables or algae wafers, and as long as the new fish isn't aggressive (aggressive bottom feeders tend to pick on mystery snails), it shouldn't be a problem. If you've got a small tank, otos (otocinclus sp? catfish) are good. Others have said that bristlenose plecos are good. Unless you have a very large tank (guessing not if you've got a
betta in there, but you never know), don't get a regular pleco, they grow to be huge.
Not sure if the betta would go after the otos, (they're pretty small), but otos are fast, and if you give them hiding places, they can survive unwanted attention. Mine have taught the fish in my tank that it's just not worth chasing after them.