Ah, Im glad to see people actually doing research before going out and buying a betta and a "betta death trap" tank.
I wouldnt reccomend using melafix, because it has stuff in it that will damage the labrynth organ(the bettas organ that allows it to breath atmosphieric oxygen), and possable kill the fish.
The best thing, in my opinion, for damaged fins would be VERY clean water, and aquerium salt(1/2 tsp. per gallon.) I would to a 100% watter change on that tank once a week. Even though it is a larger tank, it still needs to be kept clean for your little guy.
Also, if you want to put shrimp in his tank, I warn you now they may very well become betta snacks, so i would reccomend a cheeper alternative to the cherry reds, like Ghost shrimp. Granted they dont have the cool red color like the cherry reds, they are much cheeper $0.75 here where I live, so its not going to kill your wallet to replace any eaten shrimp
I'm sure he's glad you saved him from rotting in his little cup
Best of luck!