Since he's in a 1 gallon, I don't think the snail is compatible with the space given. If you upgraded your tank to 2.5 gallons or more, you could safely keep both without worry for toxicity building. Snails poop a lot, so that builds up while you're not cleaning the tank.
If his fins are getting smaller and the edges are darker/black, it usually indicates a rot. Add 1 tsp/gal premixed aquarium salt for a max of 10 days to fight the rot. Change all of his water every day. You'll want to take the snail out since I don't think salt is good for them.
After the rot is gone, change all of his water every other day.
I suggest getting a slightly larger tank for swimming space and to make changing water less frequent. 2.5 gallon tanks can be quite convenient in size for storing places and they give a good amount of swimming space. You'd only need to do 1 50% and 1 100% water change very week instead of every other day.
If you can find a heater that safely heats your tank size or if you can find a tank to upgrade to and a heater compatible with it, your betta and snail will be a lot more comfortable in water temperatures around 78-80*.
EDIT: You don't need to add aquarium salt regularly. It is actually bad for their kidneys when it is ever present and makes treatment with it less effective since bacteria and parasites can build up a resistance to it. You'll be good with just the water conditioner and tap water.