I have no windows. Venting A/C out a hinged door

I have a really odd problem. I live in a quirky, 1,200’, four story apartment that doesn’t have any windows. No, I mean zero windows; not a single window. Casement, single or double-hung, sliding, casement, bay, or awning - none. Thinking I’m making this up and I just have a weird shaped window? Incorrect, exactly zero windows. BUT. I have glass exterior doors. Endless doors. Two or three per room. The problem is they’re all hinged doors, not sliding.

I’d like to use an A/c unit on our first floor (living room/kitchen), as our single zone heat pump can’t quite reach down there and it’s consistently 82 degrees well into the night. We have. Whole fan system set up to move the cool air but it’s not enough. Okay, finally, the question:

How can I vent out my A/C unit through a regular hinged door without seriously compromising the efficiency of the A/C unit (the fabric zip-up casement things seem inefficient at best). Any ideas??