Cotton bedsheet material is plenty strong enough for the job, it's not adding strength to the bow after all, just trapping splinters from lifting up.
I do the Buckskinner/Rendezvous scene and seem to always have some lightweight cotton canvas around. I dampen it with water, run a hot iron across it but still leave it a bit damp. Then I staple one end to the bench, stretch it out, staple the second end, and use a rubber kitchen spatula to work the Titebond down into the material. Then I size the back of the bow with a bit more glue, cut the backing strips loose from the bench top, and apply it to the back of the bow. I have been experimenting with not wrapping the limbs after putting down the fabric backing and have had no problem with it lifting. I guess the wrapping with Ace bandages is necessary for my snake skin and rawhide backs that will curl and lift as they dry out.
I tore out the seat of a pair of camo pants, will have to try that for backing on a hunting weight bow. I lose enough arrows in the woods, might be nice to set down the bow and lose that for a change, ya know?