Kmartz,
The way I have used in the past and it worked well. I take a plastic cooler that just fits into my gear bag, add a roll of duct tape. Place the cooler in your gear bag and pack it full of equipment or clothes, fill it as full as possible and finish stuffing your gear bag.
When your outfitter skins your bear, fold it, I believe, fur side in and put it in your cooler leave the lid open and place it in a freezer. Remember to place your skull in with the hide and make sure it will close. When you are leaving, take the hide out of the freezer and duct tape it very, very well. Your hide will stay cold and maybe even frozen until you arrive home. Find the biggest cooler that you can fit into your gear bag. If for some reason you dont kill, just leave it packed. I learned this from a great bear outfitter in Saskatchewan, Silence of the North.
I packed a 7 ft bear hide and its skull into a cooler and it made it home still very cold. Just tape it extremely well and if you have an over night layover, which I had that trip, just ask the hotel to place it in a freezer for you. You will have a charge from the airline for an extra piece of baggage on your return trip.Good Luck and Shoot Straight.
Z