I got a couple of what I call “beach backs” from Sierra Trading. They were inexpensive and importantly fold up to a small size. The fabric is pretty funky and plasticy; I plan to replace it with some nylon Cordura or other outdoor fabric down the road. A plus is they can be situated in either direction inside the teardrop and be used outdoors at the beach!
I had the same problem. Solved it by cutting a 3/4” piece of MDF to fit propped back. Covered the edges with pipe insulation then covered the entire board with backing and then grey leatherette fabric. It is not fixed to anything, just sits on top of the mattress. As a plus it gives extra storage for two Sleeping bags and pillows. Surprisingly there is still room to stretch out in it. I’m 6’2.