Thanks everyone for your good ideas. And your bad ideas, like rodents. My wife pointed out that introducing a new species to an environment to solve a problem rarely works out in the long run.
After our failed attempts to open the door on Wednesday night, my wife had a dream that I'd somehow brought home a wacky arm-wavy guy:
So last night, after my attempts to wedge a bamboo stick into the end of a long brewing spoon so that it could be turned upright didn't produce enough force, we experimented with inflation solutions. We experimented with inflating a trash bag in the hallway. Once we got a decent seal (trash bags are known for their rather wide mouths) we noticed that the bag didn't push out with a lot of force: it would just spread or pop when faced with a heavy object. Additionally, it didn't seem like it was long enough to reach.
My wife suggested that we use an air mattress, and we found one in the closet that's about the width of the door. (The first one we considered was a double, and I was concerned that it wouldn't inflate well if it was folded on top of itself.) We spent five or ten minutes stuffing the air mattress under the door gap; the handle of the homebrew spoon came in handy again. We then started inflating the air mattress from outside the door, with the other person pushing the door open a crack.
After a decent amount of inflation, the door opened wider than a crack and I was eventually able to slip my arm in. After some wiggling I was able to gain enough movement to put a hand on the frame and push it further back. I then had my wife start deflating the mattress so I could get the door open a little further and slide my whole body into the room. Victory! I moved the frame away and we deflated the mattress to the point that I could slide it out from under the door.
I think this mechanism may have worked not because the mattress pushed the frame up so much as because it managed to put pressure on the bottom hinge such that the pin popped out and the bottom third of the door had more freedom of movement; I'd previously noted that the top of the door was able to open a bit further than the bottom, where the frame was wedged.
Some recreated photos
An amusing coda: The 15+ pound frame fell while my wife was gathering items to get rid of, after reading Marie Kondo's The Life-Changing Magic of Tidying Up: The Japanese Art of Decluttering and Organizing. However, my wife hadn't followed Kondo's instruction (based on KonMari's roots in Shintoism) to ask the house permission before getting rid of things. So the room established a defensive posture. Ironically, the air mattress that we used to regain entrance to the room is one that I've been carting around from move to move, and haven't used in close to ten years. Thanks to my "You never know when you'll need it" archivist instincts, the forces of clutter were able to save the day.