I travel in my fur coats all the time. I just ask the flight attendant to store it at the out wear closet.
If they don't have one or if it's overpacked, I flip it out so the lining is outside and fur is inside and put it nicely to the overhead compartment. I do that to my wool and cashmere coats on planes too and never had any problems.
I don't even think about it actually, it's an automatic move I do anywhere, a restaurant, a cafe, shopping, I just flip it with silk lining outside and fur inside. I also flip it when going through the security check, and just put it inside the basket. I don't use garments bags for this, it's just 30 seconds of time in that basket so I don't want to go through extra struggle.
It's especually important to flip out delicate fur like chinchilla because during flights can get easily damaged in the storage closet.