I solved the problem of not being able to access the lower cabinet while the bed is down.

1. Remove the hinges from the cabinet door.  2. Drill two 1/8” holes into the hinge-side edge of the door. I used a clamp on dowel drill guide to make sure the holes were centered and straight.  3. Cut two 3/4” long pieces of 1/8” steel dowel rod. I guess wood dowels would work too but won’t be as durable in the long term.  4. Insert the dowels into the holes that you drilled in the edge of the door. I didn’t use glue because they were a tight fit. I just tapped them in with a hammer.  5. Drill two 9/32” holes in the cabinet frame at the corresponding dowel locations.  6. Attach a small steel bracket to the bottom of the cabinet frame using a self tapping screw. This keeps the bottom of the door lined up as you slide it into place. Alternatively you could use a small piece of wood trim that runs the full length of the cabinet frame bottom.  7. Done! Now your cabinet door is removable rather than hinged, and it can be removed when the bed is up or down.

One of the great things about this modification is that it is completely invisible from the outside.  There is no visible permanent damage to the door or the frame.  If you want to put it back to stock, just remove the dowels, remove the steel bracket (or trim piece), and reinstall the hinges.