It's shipped all the time, mostly successful, but shipping meat overnight is super expensive. Several dollars per pound. Contact UPS or FedEx and ask them how much it would cost you to ship 400 pounds overnight from your address to the address of the meat processor you will be using. Costs should be the same if shipping from there to you. One concern is that it gets misrouted and ends up in a warehouse somewhere for several days unnoticed until the smell attracts someone's attention. Not sure how the insurance would cover that loss. You have to make sure it gets delivered while someone is there to receive it and get it into the freezer, not left in the sun on the doorstep to sit for who knows how long.
It's a pain, but driving home with it (and the antlers) is best. It's totally under your control at all times.
I've done it every year since 1990 (except for 3 years of tag soup) to and from Chama, NM. I'm too worried about flying with rifle, ammo, hunting clothes, then getting everything back home. A leisure drive to and from (24 hours of driving) keeps more in my control than worrying about airlines (prices and connections), vehicle rental (getting a Toyota Corolla instead of "reserved" SUV). If you have the time for travel (driving) makes it easier.