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My worries with any sort of heat/ sweat box type of scenario is massive gum damage to the remainder of the sheet.
That's an almost sure bet. I now better understand your problem. I originally thought your entire press sheet was stuck to the brown backing.
Basically, the gum in the area stuck will be irreparably damaged. It sounds like your goal is to minimize any stamp paper damage and also minimize any damage to the unstuck gum areas.
I would suggest trying a product called Stamp Lift (google it to find product info and retailer). It will result in disturbed gum, but at least it will be localized. A long time ago, I tried it for removing booklet panes from booklets with selvage intact. It did work, although it also resulted in disturbed gum in any area that was wetted. I eventually decided careful pulling/tugging of the booklet panes produced better results. But the Stamp Lift did work; I had to apply the fluid a couple of times to get it all apart. Do not be too aggressive with the stamp areas that are wetted with Stamp Lift. Remember, paper fibers get weak when wet -- you don't want to rip the stamp. You can practice with some on-paper used stamps to get the hang of it.
Just a thought.