I once had 3 Sitka Blacktails stuffed into a VW Rabbit. Of course, I was only driving a few miles, from the boat dock to my house, but they did fit. I also had a 300# black bear stuffed into it. That car was a serious piece of junk, and I was a young, single, construction worker, so bad odors in my vehicle weren't of any consequence to me.
Now, though, if I were in your shoes, I'd see if boning out a carcass is legal in your area. A boned out whitetail will easily fit in a small cooler. I'd then make sure to put the head and hide in a sturdy plastic bag. The last thing you want in your vehicle are a multitude of ticks and fleas.