I got a tube of some stuff called "Loctite Extreme Glue" from my local Morrisons. It's supposed to stick wood, ceramic, stone, glass, foam, rubber, metal, plastic and more" but - as usual - not suitable for PE, PP or PTFE.
I've used it to stick the metal hinges on my gateposts and to repair thin vinyl gloves. Seems to work OK (flexible), but don't know how it would do on your application.
It doesn't seem to work by solvent evaporation. But it does contain N-[3-[Trimethoxysilyl]propyl]ethylenediamine. So try and keep it off your skin.