This is the new stuff I have been seeing, only folks did it in the 1800's
There is a cool way to vent the room and with your situation it may work.
Folks on a recent job had a similar issue. We installed some regular doors, then made a vent for over the door. Kind of like the old windows folks used to have in thier homes which they could tilt open. But this was a couple of metal louvers painted the same color as the wall, in the header space and facing up. We did the math and it was plenty of ventilation for the room....and no vented door.
Just another option if you have room over door and no solid header. It looked nice.