A few additional points and questions but hopefully someone else chips in and gives you a second opinion.
Your tank size should be fine for a betta and a snail. Even with zero cycle, it sounds like your ammonia is rising quicker than it should for your stocking.
I cant see how stress coat can be causing your issues. I use stress coat. But perhaps get a new bottle or use a different brand of conditioner.
You say your filtration is sufficient and working fine. What filter media are you using?
Are you doing any filter maintenance?
What is your substrate? Some soil based substrate will leech ammonia into the water for a period.
What decorations are in the tank? I have heard of decs leeching nitrate into tanks, so ammonia is also possible i guess.
Have you done a good gravel vac to make sure it isnt harbouring something decomposing and adding ammonia.
What are your other water parameters? Nitrite, nitrate, pH as a minimum.
How long has your tank been running?
Maybe do an exercise so we can see how quickly your ammonia is rising.
1. Test water for ammonia. What is the reading?
2. 50% water change.
3. Retest water. What is the reading?
4. Retest again after 1 hour. What is the reading?
5. Retest 24 hours later. What is the reading?
Anyway. Onto trying to get your ammonia down. Do you know anyone who has an established tank? If so, would they let you have some established media or a sponge from their filter to put in your filter? If not, did you use any bottled bacteria like Dr Tims One and Only or API Quickstart? Products like these are hit and miss, but you need to try something. Keep up with water changes. If ammonia + nitrite combined is more than 0.5 do 25% water change. If more than 1.0 do 50% water change.