I would have to say yes, you have a fin biter =(
The habit can be tough to break.. he most likely started it in transport prior to you getting him. And stress with a new home, etc just kept it going.
Stress coat is a good thing, just don't add in a ton of it with your regular water conditioner (unless you are using the stress coat as your conditioner). If you're not, then a few drops in with the other conditioner will be fine.
I'd suggest adding in a lot more plants.. bettas tend to like medium-heavy planting, and they can get bored with little in the tank. And some members have had luck stopping it when adding in more plants. Also helps if they get to see people often, they will get lonely all by themselves if left alone too long.
Other then that, keep up with your regular water changes to help prevent rot from forming and see if adding in more plants (real or fake - here is a good cheap site to get some from here
. Plastic ones are cheaper just make sure not pointy/sharp.
Is the filter on the lowest setting or baffled? Sometimes too much current can stress them out and with long fins they feel the drag more and can bite.
Unsure what else you could do to help him.. don't add in any salt unless it forms into rot. Hopefully others who have biters can give you more ideas. Wishing you luck.