So last night I tried putting the carpet on the bed, and it actually worked. It was still kinda lumpy where the carpet would sort of bubble up where the mattress had a depression from an elbow or sitting, but once I laid down on it, it was much smoother with my eweight more evenly distributed. I didn't wake up with red rings on my ribs from the mattress springs this morning, and I actually got more than an hour or two of sleep before I would be forced to wake up.
The carpet is about 3'x4.5' or 5'. Plenty big enough to get my torso on it. That and a sleeping bag worked.