Sorry for the late followup. Here's a full picture of the door. The car has 85k on it and the hinges are not worn as far as I can tell. No play to speak of.
The car is now at Auto Sport in Shrewsbury getting aligned (I just got done putting new ball joints and bushings on the front end) and having the valves adjusted. He said he'd see what he could do, but agreed he didn't see any obvious solutions.