Personally I'd advise against a snail in that small of a tank and without live plants.. the bioload (amount of poop-increases nitrates) they cause is a bigger pain to frequently siphon out poop/reducing nitrates with water changes then just removing and cleaning fake plants/decor and wiping down the glass with a damp paper towel. In a planted tank where disturbing/uprooting plants makes it hard to put them back as they were (and some being too delicate for general algecide treatments-bleach or peroxide dips) you can better justify the need for a snail (also the plants will help with absorbing some of the nitrates they produce-faster growing plant=absorbs more nitrates)...This is just my opinion though.
Snails need a lot of food so unless you tank is COVERED in brown algae (aka diatoms) they will run out of food and need supplements such as blanched veggies and algae wafers. They can clean a algae covered tank/decor pretty quick so be ready with a food supplement when the algae runs out.
If you do get a snail keep an eye on it and the betta for several hours/days-some bettas don't tolerate tank mates and will even kill snails. Read on here that some members' bettas smashed the snails against the tank, others eat their eyes and antenna. Be ready to remove the snail if the betta is hostile.
No personal opinion on nerite vs mystery snail-I've only kept nerite so far never tried mystery. I've also kept ramshorns, mini ramshorns, Malaysian trumpet snails, pond, and bladder snails.. and I've had a few bettas who eat them.
Nerties supposedly have a bad habit of crawling out of tanks but I have 2 in an open top 55g community tank and they have never left the water (thought one got stuck in driftwood then hid on the inside black trim of a temporary tank when the 55g needed re-siliconed). I think
they'd only leave the water if quality of water is bad or they're being harassed by the tank inhabitants-trying to escape a predictor, or complete lack of food.
Another thing you need to be aware of before getting a snail: what is your water hardness? Do you have calcium in your tap? Snails need calcium for their shells or they'll break down/chip and can expose the snails innards making it an easy meal.
You can get an API liquid GH and KH test kit for fairly cheap to test your water. The GH general hardness test is mostly for calcium and magnesium (but some some other trace minerals) if takes very few drops to change color (and/or you never get those pesky calcium deposit water stains on your tank glass) you'll need a calcium supplement. Some people use cuttle bones (in tank or in filter), lime stone (small decor or crushed in bag in filter-if decor make sure betta's head cannot get stuck in small holes), or crushed coral (in bag in filter) as a natural calcium additive.
I have very hard water and a lot
of calcium-a terricotta pot with dampened soil turns mostly WHITE as the clay absorbs the tap water then dries.