As stated above, hit and miss. Even when all has been harmonious in a tank, even for months, one morning you could wake up to disappeared shrimp, fish snails or what have you. Some times they go crazy on tank mates for no apparent reason. Have heard this same story time and time again, and had my Ghosties disappear too, before I was here at AA.
The Ghost shrimp are a healthy snack for the Betta so it won't hurt him, but it could make you sad. I personally love shrimp as aquatic pets, but it is also ok to me if some become part of the food chain. Not that I prefer critters I think of as pets to be eaten by other pets
. Just so you can be mentally prepared, and or prepare the kids if any as to the process of the food chain.
I would try a few Ramshorn snails or one Mystery snail. I would say several Mystery snails because they are fun to watch. BUT if one very large grown up one dies and you don't notice, it can really foul your water, and endanger your fish, and if several die at the same time, let's say a strange water pH change or something, BIG problem. I have had this happen. If you are planning for a larger tank in the next 6 months or so, then you could start out raising some baby Mystery snails and move them to the new tank later.
Some Bettas hate anything in their space. Even a little snail!