Could be columnaris or just a fungal infection growing over a wound. "The fungus spores can cause infections when the fish already has damaged gills or skin. They are typical secondary infections. White, cottony growth appears on the body of the fish. Anti-fungal medicines fight the infection."...
I did that with my tank, although I would recommend using most of the water. I set up a 45 gallon using RO water and the water from the 20 gallon. I set it up with a new filter but added old filter media from the 20, once the RO water was warm I started adding the water from the 20 gallon, then...
Bettas like warm water. I keep mine around 80F. When you change the water, make sure you use a water conditioner and do not wash with soap as a friend of mine was doing! I have mine in a 5 gallon with heater and filter and I never change more than half the water at one time.