Please do not ever, ever freeze a fish to death; it is a very painful experience, and is indeed quite cruel. It only looks peaceful because the animal is slowly being paralyzed by ice crystals forming in its tissues and organs. And, since this is a betta, it would not just suffocate before freezing occured.
Clove oil is the most humane way to euthanize a fish. I do not know exact measurements either, but when I euthanized my goldfish - who also had dropsy, and was taking long, slow days to die - I basically scooped him into a cup so that it was filled half-way with water, then took water from his tank into a bottle, added a TON of clove oil, and shook it until it was well dispersed, after which I poured it in to fill the rest of the cup. The fish was completely out in under two seconds, and died shortly thereafter. However, to make sure it is def. dead, I left him in the cup until his eyes glazed and slimecoat shed. Then I froze him while he was still under the effects to make extra sure he'd never wake up.
Clove oil contains an anesthetic, so your fish will not feel any pain. It is essentially like euthanizing.
Anyways, between the white fuzz around the gills and heads, the fin deterioration, and rapid onset of the illness (which has now gone so far as to be Dropsy) - as well as the fact that it seems to be contageous - I would make a strong guess at Columnaris. If you scroll down this board, I have a large thread on treating it, but sad enough to say, it is probably too late. I think the best you could probably do for your boy right now is give him a peaceful death.
I'm so very sorry.
edit: wow, like... 50 people just responded in the time it took me to post that, and said everything I did. I feel stupid.