Ive used carpet too. I was thinking of useing the moss in bags and then using them around the house when I get tired of looking at them. You could try and cover a piece of OSB board with a shock absorbing rubber.

You could built a stop out of 1/2" plywood 4'x6'x2'.Use insulation board 2' foot thick to fill the center of it. White is the cheapest. Then cutout wholes to recieve 2 cube targets one on top of the other. Cover the back with a sheet of exterior 1/2" plywood for rigidity. Use 2x4's in the corners. exterior glue an screwed. Give it a good coat of paint to protect it and make the wife and neighbors happy. Put wheeels on one side and tip it towards you to move it around.
The plywood cdx 3 sheets should be around $45. the foam depending on the type 2" thick (white $2 a piece, blue$25, use the white). Wheels could be off of an old bike or lawn mower if not $10 a piece. Cubes ?. It would be fairly light weight so you may have to anchor it.