I've had this same thing years ago, and it kills quickly without significant meds, and even then it's iffy. Throw out *everything*. I kept trying to clean everything, and just ended up killing lots of attempted rescue bettas. They had the same symptoms, same rapid death, and it's almost impossible to get out of everything without some serious cleaning/sterilization. If you have a filter, unless it's a fancy filter, I'd toss it too. If you keep the filter, make sure that you get rid of everything that's NOT hard plastic in it, as any sponges or mesh can harbor the fungus and will be next to impossible to get fully clean.
I finally ended up getting rid of all gravel (switched over to marbles because they can be boiled), ran everything through a dishwasher heavy cycle with detergent, then a second one without detergent to rinse at very high temps, and then I set the tank outside for about a month to *bake* in the sun. Only then did I reuse the tank without having this same thing strike again, and again, and again.
If you have decorations that you hate to part with, boil them. Boil them until a good deal of the water evaporates. You are effectively sterilizing your tank with this, it's similiar to what autoclaves do to sterile instruments for surgery.
This is the only thing that worked for me. This thing doesn't really follow the traditional rules for fungus, so I'm not sure if it really is one, but whatever it is, it's bad, and it's been in the betta community for the last 13-14 years, because that was the last time I kept having trouble with it.
I know that people say that you need to keep some bacteria in the tanks, but with this, you've got to kill everything to make sure it doesn't come back. Just as long as you clean it all thoroughly, to the point that nothing, not even fungal spores, can survive, then you'll be fine. :) You can always restart with fresh, good bacteria. It's not worth having whatever this is in there just waiting around.
I kept Jungle Fungus Eliminator on hand, and if any of my guys got it (it was coming from the pet stores, not just one, several of them in my area), then I'd *immediately* start the Jungle treatment and they usually survived. If they're strong to begin with, then they're usually okay.
I'm so sorry about your fish, and I don't mean to scare anyone. It's not horrible to get rid of if you just make sure that everything is *really, really* clean, and then you're fine. :) It's just usually not worth it to clean anything except the tank. :)
Good luck to you, just clean that sucker to within an inch of it's life, and you'll be safely ready for new fish. :)