I think that I might try to sneak some glue under the skin that has raised and then clamp that down somehow. Maybe with tape would be good enough. You could probably use any glue for that. I don’t think that the trad rice glue is necessary. Even CA (super glue, perhaps a thicker consistency) would do the job. True, if the skin is ever replaced that glued bit wouldn’t come off like the rest but that small area could easily be scraped down to allow a new skin to be glued on.
I live in a very humid part of the world and have a few thoughts on dealing with heavy humidity. (N. Thailand in the mountains/forest and often it is basically a rainforest sort of environment.) It might good to find some place that is dryer than elsewhere in the house. (I have a workshop with a dehumidifier but that’s a different story) I don’t think that extra layers of fabric/towels will necessarily help at all, as those materials might tend to absorb and hold moisture rather repel it.
One thing that I’ve done in really humid times (and when my dehumidifier bit the dust a few months ago! I have a new industrial one now) is wrap instruments with a plastic cover of some sort, maybe over the case, and then add some sort of desiccant thing such as silica packets inside. You could also pick up some of those cheap digital hygrometers to keep on eye on the humidity level. They’re not perfectly accurate but would give you an idea of how things are at least.
Just a few thoughts. The same thing happened to me on one of my shamisens when i first got here 4 years ago.