A nerite or two wouldn't hurt as a tank mate in a 5 gal. IF they are pond snails they can be considered a pest. Not sure how they would affect the bioload, I've tried to avoid them, however I managed to get one... very large one.
The only other smallish snails I can think of are smaller trumpet snails. They will breed prolificly depending on the amount of food.
None should hurt your betta, he might hurt them, some have been known to make snacks of snails. The only other thing to add, is mystery (aka apple) snails are poopers. HUGE. I have three in my 33 gal. If not for the live plants it would be dieing from the ammonia they have to be putting out.
Side note, snails need food, if you have live plants they will mostly take care of themselves (not necessarily in any way you want them too). Algae, algae wafers, or veggies.
Hope this helps, and gives you a bit more information.