Plan is to pick up some materials later today and mock things up. The metal on the base provides the base for the seat and a place for the hog rings to attach.
I'll look again, but not sure that riveting would do it, the who bit along the bottom is about to turn to dust. I just took a picture of the worst area.
Still thinking a piece of metal rod along the bottom for a solid base, and then an inverted L-shape bit of sheet metal tagged to that for the ring attachment holes.
I'll need to separate the vertical frame parts from the rusted area. I think a cutting wheel or multi-tool should be able to do that.
Before I remove any of the rusted stuff, I'll mock up what I think is the answer and see if it makes sense in reality.