You have a pond snail. If you have one, you'll probably have more very soon! You shouldn't have any problem keeping it alive because they seem indestructible and multiply like crazy unless you don't allow uneaten food to stay in the tank. I do only have 2 in my 10 gallon... they don't multiply because there's no extra food. I feed my bettas one pellet at a time. They do eat algae, so it should be fine in your tank if you have some algae. I've heard that if there is no algae they will eat live plants. Sometimes they will eat dying leaves no matter if there's algae or not.
Your betta may eat the snail. My female bettas put a huge dent in my snail overpopulation in my 55 gallon tank. But my male bettas don't eat the ones in my 10 gallon tank.
Pond snails seem to be okay with all kinds of parameters and temperatures. I actually have pond snails that have survived 90 degree weather. I had a plant tank for quarantining plants before putting them in my big tank, and anyway, in that small tank I ended up with tons of pond snails, malaysian trumpet snails, copepods, and these REALLY CREEPY worms that lived in the gravel and would wave around slowly and creepily. They freaked me out so much that I just moved the MTS to my big tank, and took the entire plant tank outside and left it there on my back deck. That was in the spring, so it's been through weather in the 40s all the way up to the upper 90's, and when I looked in there last week there were still about 5 or 6 snails crawling on the glass. I also have not done a water change since I put it out there, obviously, and they're still alive. So they're kind of indestructible.