When I was a kid my uncle had something similar with an old army canvas tarp, the olive colored heavy ones. The range was in his basement and it was held up from the floor joists. It worked great for my youth bow but can’t imagine that it would hold up to something greater that 40lbs or so.
I like the idea of the sheet rubber or maybe even rubber roofing if you could find some used stuff. Separating two layers with a small space in between I would imagine would stop most reasonable compound speeds. Not sure how hard it would be on arrows though.
Good luck. Would love to see what you come up with you are done.