I have nothing to add here (I use a ultralight thermarest wrapped around a very light weight sleeping bag but have a new idea inspired by the squares mentioned above.
How about cutting a roll mat into triangles to put on one side of your bike. Could tape them together to create hinges to stop them spreading out in the night (or even better, eyelet them and tie them into correct arrangement) . Attached to one side of the frame via velcro loops/paracord you could still access bottles/frame bag on the other side.
Edit – found an old army issue roll mat on the hill the other week. I’m off to flatten it and get the eyelet tool out…