It will easily fit in the cooler, even with skull and wrists/paws/claws intact...or hide and skull seperated. I would assume that the hide will be sans skull, with most of the meat fleshed off the hide? It's easy to do if you have a couple of hours to kill on the backside of the hunt.
Just a thought: I have shipped meat and hides in some of the waxed cardboard boxes many ship fish back from here. They are durable, much less expensive than coolers, and will easily pack a hide (they come in several sizes.)
Have a good trip and be safe...
bhtr