Fin Rot can usually be healed pretty well with Aquarium Salt and clean water. Just QT him and use 1-2tsp/gal Aquarium Salt. Perform daily 100% water changes. Don't use the salt for more than 10 days. If you don't see any improvement after 5 days or so, then you can use Tetracycline as well, but don't rush to medications. Overuse of medications leads to drug resistance.
Never use Melafix with bettas or other labyrinth fish (e.g. gouramis). It contains a tree oil that harms their labyrinth organ.
Penicillin is 250mg/10gal. Repeat treatment every 24 hours for no more than 10 days. Perform daily 100% water changes between treatments. Ampicillin has the same treatment routine. Both Penicillin and Ampicillin have lost effectiveness over the years due to overuse. Many bacteria are resistant to them. Aquarium Salt should be all you need.