This mattress that I got for it's portability. It folds into thirds which makes it easy to transport in my CR-V with all my other stuff.

Since you mentioned folding a section of the bed, this could potentially be a good option as it's designed to fold itself. Although I'm not sure if they make a bigger version that a twin or not.

This mattress seems similar in concept though, and comes in full size. And the fold pattern actually looks like it might be better too for your needs. But of course measure first.

I can't speak for the second mattress I linked, but the first one is very comfortable despite only being 3" thick. I'd guess the second one is good too.

Best of luck, and I'd definitely love to see what you come up with!