wow it a big difference. How long he is like that . Go buy unscented aquarium salt at the pet store. Get him hospital tank or you can keep him in the smaller container (you can use the one you bought him in) you definitely need to do aquarium salt and if he is not better i guess medications. I would really get his smaller tank, do salt treatment , no filter he needs less water movement.
You will need to pre mix 2tsp/gall make sure it dissolved. It better if you can pre mix it in one gall jug. Shake it well. Do daily 100% water changes. You can do it for a few days and you might need to switch to 3tsp/gall.
Make sure you acclimate betta to the new water temp/chemistry. You can use salt up to 14 days, but if in 3-5 days he will not get better or will get worse you will need med's. How long he is like that, haw fast his fins melting down?
Elsewhere i didn' see your response ... I think it looks more like fin rot