Ok, here's what I discovered last night.... I tried 3 different pieces of material: plywood, Masonite and card board.It stuck to all three and almost felt like working with fibre glass. This morning I went out to try to peel them off the pieces. It put up a fight and the parts that weren't glued down separated from the rest of the cloth with about the same effort you would rip a blue rag with....maybe a bit more but what was glued to each material stayed put. I took and tried to rip the card board in half, and it tore pretty easily until I reached the fabric and glue and it got really tough to tare as compared to just card board or just fabric. I'm impressed with that so I'm thinking multiple layers could POSSIBLY do it.. I also like the texture of the cloth as it's fine. once it was glued on while still glossy and wet it reminded me of denim.
I'm still not sure about this, and am still leaning towards canvas for the strength but I think this weed fabric has potential for some other little projects I have in mind. Mabey I should do a 4 layer experiment eh?
As for using any wood glue? I believe your rite but happen to have tight bond out there from another project left over and absolutely love the stuff.

I've used Elmer's as well and have had good luck but as I recall standing in Lowe's TB was less expensive for the size I wanted to buy
Hope my rambling made some sense!