This is what's on DM's sticky for treatment:
Tail rot or fin rot
•Symptoms: Betta’s fins and/or tail seem to be getting shorter and shorter or they seem to be falling apart and dissolving, Black or red along the edge of the betta’s fins/tail, Bloody tips, Behavior may not change
Treatment: Conservative: Treat with Aq.Salt at 1 tsp/gal. Increase water changes to 100% daily. Replace accurate amount of salt following water changes. Add Stress Coat to help repair tissue. If there is little to no improvement within the first 5 days, you can increase the salt dosage gradually to 2tsp/gal but do not continue any salt treatments past 10 days. Medication: If Conservative treatment is ineffective use API Tetracycline, API Fungus Cure, API Triple Sulfa, OR API Erythromycin. Also add Stress Coat to help regrowth. Continue until fins/tail stop receding and start showing some new growth.
However, I still don't know exactly how severe your betta is suffering from the fin melt or fin rot, and since your betta has been having it for a while, I would just try the conservative way first with AQ/IAL/StressCoat on daily water change to see if you notice any improvement. If after 10-14 days and no signs of improvement, I would go for meds intervention. I researched and it seems that the tetracycline or the Triple Sulfa are the drugs of choice. I guess it also depends what you can get your hands on too.
Good luck and keep us posted.