For what it's worth...
I usually fly thru Phoenix where they have required the cape to have been frozen and rethawed for inspection and I have been thru there with multiple animals on the same trip without incident. I went thru Dallas on my way home this year. The TSA guy copied my paperwork for the fish and game dept., washed my hunting boot soles, advised me that if I had multiple animals that I would be required to take them to an "approved establishment". I think he meant taxidermist that was registered as such. Anyway, he never once looked at the cape even after I asked if he would like to do so. It was a totally different experience than Phoenix but much easier. I've heard to stay away from LA!