I've used carpet "samples" layered for use as a BH target. Shoot the arrow, peel back the layers, remove the BH, then pull the arrow out. My target is made of 18 samples freely stacked together, but in contact with each other, yet not compressed or bound. My arrow speed is only ~240 fps & I get a penetration of ~ 15 layers. You may need more. After a while, say a couple dozen shots in the same area, it gets a bit ratty. Spot shooting will give you better life.
As for using carpet padding instead of carpeting, that sounds like a good thing to try. I'd really be interested in the results you get. Please advise.