In a 3 gallon Eclipse I would just go ahead and do huge water changes with temperature matched water. Try to get the temp down to about 76 degrees, but slowly. Bettas like about 78 degrees but so do bacteria at 77 degrees and higher. Maybe you could put a fan near the tank to blow on it to cool it down.
I wonder why it's so warm... you're in Maine. You must keep your house pretty warm? (The temperature fluctuates extremely in my house, especially upstairs. It's warm in the day time but freezing at night. The tank seems to stay at 80* or a little higher. I'll try to keep it down around 76. I keep the cover propped open with the light still on until I go to bed or go out somewhere then I have to close it, but from now on will keep the light off until I can get him cured because that is probably heating the water also. The problem with leaving the lid completely off is I have 2 very curious kittens that love to stick their paws in my tanks and as of the moment I have nowhere to put the 3gallon where they cannot get at it. The screen thing sounds like a good idea but I don't really know how I could keep it on there without having them knock it off or try to stand on top of it and fall threw like they did with my bearded dragons cage.) Don't use water changes to bring the temp down, just let it cool down slowly with a fan blowing across the top of the tank. If you could leave the lid off, but cover it with a screen so he can't jump out that would help.
I have a Brochis in a 5 gallon Eclipse and have been doing 90% water changes daily to treat finrot. His fins look almost 100% after about 4 or 5 days. I use Prime and match the temp. I've also been using Melafix, but I don't know if that is good for Bettas. I've heard many people say they use it on Bettas, and many people say not to use it, so I don't know for sure.
I would do daily big water changes, as much as 80-90%. AmQuel is good, but so is Prime. Many people that breed and raise Bettas and keep them in small containers or tanks do daily 100% water changes, but I don't think that's necessary. I wouldn't do the salt. If his water is pristine his fins will probably heal in a few days. (I will definitely be upping my water change to 90% with temperature matched water. I had been doing about 25% daily as I wasn't sure if it would shock him or not. I use Prime for a water conditioner.)
The tank can still cycle even with big water changes because there will be enough traces of ammonia in the tank, even if it is undectable when tested. The clean water, however, is of utmost importance to cure finrot. I would just think of the 3 gallon tank as a big bowl and do big water changes. Keep the filter in it and eventually it will have a good colony of beneficial bacteria. (That's the problem it should already have tons of it from the tank I stole the media from. Which is why I'm not quite understanding why when I test it nothing seems to show. I must have messed up somehow and killed my bacteria.)
I like having a bubbler in a tank to aid in oxygenation for the fish's comfort. Even though Bettas are labyrinth fish and can breath air they still use their gills, too.
You're using the liquid test kit? (API Liquid Test Kit) That's much better than strips.