The good news is, AZ is not a CWD state so you don't have to worry about those requirements. Check AZ, but every state I'm aware of does not require evidence of sex post-processing. The tag/carcass coupon accompanies the meat. If you haven't caped an animal before and decide to go the shoulder mount route, I strongly advise having an AZ taxi cape it off the skull and flesh the cape for you. The ones I use don't want it salted, just fresh frozen. If you aren't going to do a euro, take the time to flesh and clean the skull cap yourself. If you are going to do a euro, time permitting you should do the initial boil off down there. Otherwise, freeze the skull.
Southwest rules on perishable luggage cover harvested fish, fowl and game:
https://www.southwest.com/html/customer-service/baggage/special-luggage-pol.html . This would also apply to a frozen skull for a euro. Purchase a reusable cooler (not Styrofoam) in AZ big enough for the meat and antlers/skull/cape, a sharpie and duct tape, and follow the Southwest rules for perishable luggage. antlers and cape in first, then pack around with frozen packages of meat. Tag goes in last on top of the meat, keep the rest of the license/permit with you. Latch the cooler, then wrap around 3-4 times at each 1/3 of the cooler. Use the sharpie to write your name, address and phone number directly on the cooler - a white lid is handy when cooler shopping. This will likely cost you $75 for an overweight/oversize checked bag, so feel free to top up the cooler with boots, dirty clothes, etc (suitably bagged). Keep it under the 100 pound max weight. According to Southwest's website, they don't prohibit any items beyond the TSA list.
I think that will be good to go, but certainly run it by a Southwest representative by phone before you go.
