I might also advise doing the same for any other states you transit on the way home, just in case you get pulled over.
This brings up a couple of thoughts about the best way to carry a deer back without running the risk of causing any troubles in the "in between" states. I would think that it would be easier carrying processed meat than a whole animal. If I were having to carry either a whole animal (or a head for mounting), I think I would try to get hold of the game warden in the state where it was harvested and get him to "certify" the animal as being taken there. That way, any other state would not have reason to suspect the deer was harvested illegally in their state.