When I saw your door I thought maybe you could pull the curtain to the other small window instead of the 6 inches on the other side of the door. It might stay out of the way better. Also, I realize you have a pet door there but maybe the quilt wouldn't come all the way to the floor. You could make a matching dor thingy to catch the drafts at the bottom of the door.
Another suggest is getting one of those swinging rods that you can just swing it out of the way. Make the back side as pretty as the front and enjoy a different quilt in the day time!.. I will be interested to see what you come up with.
Good luck!!!
Sharon from Maine