Without seeing it, the best solution I can think of is to try cleaning it up real good with something like Simple Green(works amazing on rubber). After it’s nice and dry, use some Eternabond RoofSeal tape making sure to have at least two inches overlap. That stuff will stick well to clean rubber and hold good for a few more years. Nothing will ever be a permanent solution until you can replace the whole piece of rubber roofing unfortunately .
I don’t have anything specialized like simple green, but wouldn’t acetone or denatured alcohol make a decent primer? I think priming the smooth rubber surface should be the easy part but it’s the bitumen band that’s tricky. It has a sandy texture and sand crumbles off when I rub it. Perhaps I should steel brush it where it needs adhesive.