I couldnt beging to even think how many deer and hogs I have killed and transported. Temps vary wildy during our hunting season, and it has never been an issue.
Hogs in the summer we just quarter and throw in a cooler with ice. Deer in early season, we just quarter and throw in a cooler with ice. Hogs and deer when the temps are 45F and lower, we just field dress and throw in the truck bed. I guess if I didnt drive a truck, transporting could be a little tricky, but I think I would just get a decent tarp, and throw it in the trunk of the car. If I was adding ice, I would use the 10 pound bags, and put one or two in the body cavity.
Not sure on the level of skill your dad has, but I can normally skin and quarter a deer in about 20 minutes. Hogs take me longer because I think they are just filthy critters.
Either way, usually most folks are not moving one for any long length of time, and I would believe a field dressed deer with 2 x 10pound bags of ice in the cavity, wrapped in a tarp should pretty much cover anything. If the deer is not field dressed, well that is a whole new issue.