OK, I think I figured this out. First I tried to use a cheap caribiner to keep the supports from bowing out.Ronin said:I see what you mean - that would be concerning. Maybe an extra leg centered opposite the 2 hinges to take the pressure off the hinge. You could use another folding hinge attached to a piece of 1x1 stock. Good luck!
That didn't work so well, as with the supports actually narrowing more than they were supposed to, there were no opposing forces so both supports bent in the same direction (staying parallel) and losing even more rigidity. No pic of that, sorry.
So I contemplated some kind of supports for the seats, like Ronin suggested above. I couldn't see how to do that without making the seats unable to fold away flush.
OK, so how to make them stronger? I pondered putting a chunk of wood between them, but found a section of C-channel that fit around them instead. Here is one in place:
With it in place, there is no way for the supports to bow out, and the C-channel fit snug, but not so much that the supports bend inward either. Here is are a few pics of the shape I cut, and the size I used: