Housing
What size is your tank? 5 gallons.
What temperature is your tank? 81F
Does your tank have a filter? Yes.
Does your tank have an air stone or other type of aeration? Yes.
Is your tank heated? Yes.
What tank mates does your betta fish live with? None, just plants.
Food
What type of food do you feed your betta fish? Hikari Betta Bio Gold pellets
How often do you feed your betta fish? Twice daily per food package instructions.
Maintenance
How often do you perform a water change? Weekly.
What percentage of the water do you change when you perform a water change? 20-30%
What type of additives do you add to the water when you perform a water change? API Betta Water Conditioner and API Stress Coat +.
Water Parameters:
Have you tested your water? If so, what are the following parameters?
Ammonia:
Nitrite: 0.25ppm
Nitrate: about 2.5ppm
pH: 7.6
Hardness: Not tested but my water is relatively soft.
Alkalinity:
Symptoms and Treatment
How has your betta fish's appearance changed? His tail fin has grown shorter and ragged, and the red spot on his top fin has grown from a tiny dot to a small patch. These changes have taken place over about 3 weeks.
How has your betta fish's behavior changed? It hasn't.
When did you start noticing the symptoms? These symptoms have taken place over the 3 weeks I've owned him, I've been saying "huh, maybe his tail is a little shorter" for about 2 weeks.
Have you started treating your fish? If so, how? I have been treating with salt for about 4 days but it has gotten slightly worse.
Does your fish have any history of being ill? When I bought him he had beautifully shaped fins and the red spot was there as a tiny dot that for all I knew belonged there. Things have just gone downhill from there.

How old is your fish (approximately)? Beats me I bought him at Petco 3 weeks ago.
I can't quite figure if he has fin rot or is biting it or both. The edge of the tail fin is not raw or red at all, it's just shorter and ragged, which sounds like biting, except I've never actually seen him biting it, and this doesn't explain the red spot on the top fin.
Photo attached.
I'm thinking about going out to buy him some medicine tomorrow if I don't get any better advice. Treatment with salt is not helping.
This guy is my favorite fish, I really want to save him, please advise.