melafix is not a good med to be used with bettas as the melacula (spelling?) oil in it is not good for labyrinth fish, which a betta is. it has something to do with the blocking of oxygen uptake into the fish's lung organ, from what i have read. there is a product out there called Betta Fix and it too has melacula oil in it (albeit a lower concentration), even though it is marketed to be used for bettas. i once used Betta Fix and i believe this ingredient is what caused the demise of my betta.
you may have better luck with a product called "betta revive" by Hikari. i used this exclusively when i had bettas for fin rot when it did happen to occur. worked very well. it does have methylyne blue and malachite green in it, so it will stain silicone and any softer plastic and fabric plants. suctions cups in the tank will forever be blue afterwards, lol.
white bettas, i have read, are not as easy to keep as other colours. don't know why this is but it might have something to do with the lack of pigmentation?
the fin rot bacteria may have taken such a hold that it will be very very difficult to cure no matter what you do. it has been noted that if the rot extends to the body there is a very slight chance for recovery.
i would do daily 100% water changes on his tank, remove any substrate (could be harbouring the fin rot bacteria), clean the filter well in tank water that you have taken out during a water change, not under running tapwater, as to preserve the good bacteria that is present in filter media. might be a good idea to sterlize everything if these steps do not help. you can sterilize plastic and fabric plants by boiling them, and this will eradicate most yucky organisms and such. mind you, sterilizing everything, including the filter, will cause you to have a brand new tank after, so ammonia and nitrite levels will need to be monitored until the tank cycles again. large volume water changes will be required on pretty much a daily basis to keep both the ammonia and nitrites at zero, to keep the fish from getting ill or ultimately passing away due to ammonia or nitrite poisoning.
hope that helped! good luck with your lil finkid and keep us posted on his progress!