OFL got here before I did.. she may have more info then I do, but from what I gathered by reading, it looks like a few things could be changed right now to try to help him immediately. Once we get his strength back, then we can work on the rot.
Gradually get him acclimated to no Melafix- cup him in the cup you got him in, or another small cup (with the water with the medication in it if he is still in there) and over a span of 20-30 minutes, remove a portion of the medicated water from the cup and replace it with clean water (empty the 2gal tank he is in with medication and fill it back up with tap water of the appropriate temp of 78-80*, and water conditioner).. keep removing some of the cup water and replacing it with the new clean water about every 5-10 minutes.
After the time limit then use a net and gently pour the cup water through the net, gently netting him and place him in the tank.
So he should be in just plain tap water with nothing added but conditioner.
Let him sit in there for at least 24 hours to help clear out his system.
After the 24 hours, let us know how he is doing.. how his color is, how his actions are, etc.
Fin rot tends not to cause a behavioral difference in them unless it has reached their body- which then you would have to go to a strong medication. I rather not jump to those just yet, but to get a better ideal on what is going on with him, I want to see how he is outside of the melfix- as a lot of medications can be harmful to their internal organs, and I want to rule out medication reaction vs an underlying issue other then the fin rot.
As OFL asked, length he was in the AQ salt.. as the amount you used really wasn't enough to do anything for the bacteria, but don't want to put him back in it if he was in there for a while, as even AQ salt has potential to cause health issues if over exposed.
You say 25-30% weekly water changes for the 10 gallon- are there live plants? If not, you may want to up it a little..
You said added "materials" for chemicals you put into the tank, along with conditioner- what is the material, and what conditioner are you using?
What sort of catfish is it?
Neons tend to be fin nippers- and with only 3, they may not feel secure with such a small number and may be after your betta, and not the betta taking a toll on his own tail.
Only feeding one pellet a day is a very very small amount, especially for a fish that is commonly called pigs. What type of pellets are you feeding him, and are you sure he isn't eating the tetra's food? I would guess no, as he looks a little on the thin side.. I would suggest getting a higher quality brand that tends to be a favorite among picky eaters- Omega One Betta Buffet, or the New Life Spectrum betta food..
-- so to recap- get him into clean water, slowly acclimating him to it.. at this point, I rather risk him leaving mid-treatment then to leave him in there.
Oldfishlady may have some other ideas on how to go about it- if so, then I would follow her wisdom :)