That made me laugh so much Asa! For my floopy bags I use a Purseket. It kind of props them up and keeps them from totally collapsing on themselves. I don't have a Birkin but I think it might help.
I say leave them just as they are while using them (or toss in a Purseket to help organize your belongings and add some structure) and consider doing what I do with mine when they're not in use. Stuff a beautifully covered travel pillow inside! Works like a charm!