Now this coat is totally clean and wearable! I took off the fur trim and then bathed it in the bathtub using Woolite and really hot water three times. Then I put it in the sun on the back porch to dry.
Reattaching the fur was a giant pain in the butt, so let's just say this coat is never going to be cleaned again. It took an entire Poirot movie and an episode of Project Runway. That's like at least 2 hours. I also repaired a tear in the back lining and a place where the hem was coming undone.
I then sprayed it with Febreeze and let it air out on the back porch for about a week. It no longer smells like BO or has dirty tissues in the pockets, but it is slightly musty with that old house smell. I think it will air out with use, although it isn't cold enough for a wool coat yet.