I think I'm going to have to attempt Sewing.
This seems to have been moderately successful. I've fashioned a new seat out of medium-density cotton duck (tradeoff of tensile strength for ease of sewing and lower frayability), attached it to the frame and not yet given myself a horrible injury by bouncing and squirming around on it.
As I was reluctant to drill out the rivets, I sewed it to the holdy bits in situ. Have you ever tried to feed fabric through a sewing machine with a stool attached to it? Not easy. As a result it looks mildly wonky (mostly down to the final leg not being as neat as the others because I couldn't see what I was doing), but that's not something you notice when sitting on it.
Will see how it goes. If it doesn't show obvious signs of falling to bits after a field test, I might have another go, hand-stitching the leg attachments.