This sounds like an all too common bacterial infection. I have lost 6 bettas to the same disease all within 3 weeks of the first one showing the symptoms. In all cases, I failed to stop the progress of this infection and all fish were dead within 12-24 hours of the first sign of something being wrong.
In all cases the first sign was a small grey patch, which looked as you say like either a scale raised or peeling. The patch then spreads and in 4 or the 6 the fins rotted away as well. This is a particularly nasty infection, and it seems to only effect bettas as 2 of my 6 were in community tanks and nobody else suffered any ill effects.
Meds I tried were:
maracyn1 and 2 together.
Oxytetracylcine.
Nifupirinol
All to no avail. I'm truly sorry to hear about your fish, but I am trying to build a pattern for this, can you tell me if you kept your betta with salt in the water. I have so far found no patern, this isnt casued by poor water quality, poor diet, stress, or old age, as none of these are present in my betta tanks, so I'm still hunting of the casue of this illness myself.
The only think I can suggest at this stage is to completely break the tank down and sterilize everything, what ever this bacteria is its a betta killer, so dont add another betta to that tank till its fully 100 percent nuked.
One of the main thing that comes back time and time again, is the poor immune system modern bettas have, this could be inpart due to large betta farms pumping them full of growth enhancers and oxytetracline to prevent illness setting in, in the filthy puddle they are bred and raised, it could also be due to how lose to the gene pool bettas are bred, another fish thats starting showiung similar symptoms is the very popular dwarf gourami. Its with this reason that my local fishstore have taken both fish off their lists. I am currently betta less and in search of a small local breeder to see if this assists in reducing this problem
Again sorry to hear about your betta
RIP shyrome