I've replaced the latch, so it is pretty much staying shut, although I'm still using a bungee, just to be safe. I might restore the hinges too eventually. They are a bit funky.

 The bottom of the door is so loose that I can see daylight and stick my fingers through without pressing. Scamp sells replacement doors, but I ain't paying that much. I am engineering a system to put the bend back in with four pieces of oak, and a fifth to warp it back into shape. When I get it started, I'll post pics..