Instead of a pillow, maybe use foam cut to the approximate size of a folding chair seat?
Make a pocket/case for the foam - and then add "saddle bags" to the sides of the center. Perhaps add some of that non-skid stuff used in cupboards/cabinets on the bottom so it stays put on the chair.
It would end up being a rectangle. You could add ties or straps for added "security"
I don't have any ideas for the backs.
I have no ideas for