I don't know how long your bungee cord is, but they now make bungee cords with carabiners on each end. I'm not sure if they make them short enough for your application, though.There is a hook and eye latch: one to keep it shut and 1 to keep it open. There isn't that spring thing to keep the hook secured though. I'd been pondering ways to do that. I'm going to have to check those out (after looking more closely at the current set up for modification points.
The hook to keep it open is actually a black rubber bungee cord with the hook on one end replaced with some heavy wire, making a hook about 2 inches long. Somehow they're getting that out of the tiny little eye on the door..... hmmm....maybe I can contrive another piece that slides onto the hook after it goes through the eye.
