If you can find it, dry ice is the answer. Pop some in a paper bag and jam it inside the body cavity. It will cool down the deer (or bear) and yet won't make a wet mess all over the place. Handle it with gloves. It will last quite a while in a good cooler. You can find it for sale at many larger groceries, welding supply places or dairies that do mail order (for sending ice cream through the mail.)

Chubber