Firstly, it's just torn fins - NO salt nor medication is going to help him - can actually harm him. Melafix, if not VERY careful can easily kill the betta.. use only in extreme cases and really diluted. AQ salt won't help regrow the fins.
The only thing that is needed is just clean water and good food.
2 100% water changes a week in a filtered tank is too much - there needs to be a balance of multiple things to keep the tank and fish healthy. There is such a thing as "too clean/sterile".
A 1 gallon does not need a filter, bettas do not need the aeration from the filter.
Options are - take the filter out and do 1 50% and 1 100% water change per week. Or - keep the filter in and do 2 50% water changes per week (1 includes siphoning the substrate). The important factor here is cleaning the substrate each week (if you have no live plants) - as the filter doesn't remove waste, just hides it.
Do one of those two recommended water changes (with or without the filter) and he will be just fine. Add in salt you can potentially hamper the new growth and harm his kidneys; salt should only be used when it's absolutely needed. Using medications can also be detrimental to their organs if not very careful - Melafix/Bettafix does have an ingredient that has been said to cover their labyrinth organ and cause them to suffocate - why it's best to avoid it all together, and if you do use it to dilute it.
You can do too much and go overboard when caring for these fish.. there is no need to do anything other than keep their water clean (but not sterile) when you have a rip in the fin. The fin will grow back and repair itself on it's own.. why put the fish's health in danger?