I'm a long-time clothing seamstress. I consider fleece just above handi-wipes: Semi-disposable. It's junk best left for souvenir sweatshirts. It will be full of holes before you get it all glued on. It pills, it tears, it's uneven. There's no weaving going on. There's no thread. It's crap. You will regret the attempt immediately.
I'm not sure what else you could use, though. Donna's right about thinking about flame resistant materials. Casita
may be right about using some kind of carpet. I don't know if theirs has a woven backing, or rubber. I'd probably stick as close to what they use as possible.
Let us see some pics when it's done!