Having done this once, I can offer my suggestions, but it does matter where you are flying to and from and the size of the airplanes that they have there. Bethel was just able to handle what we gave them on the way out. It had to go through Seattle and end up in Kansas City rather than Omaha or Des Moines where it would have been more convenient for us, because the planes at these destinations were not big enough to handle packages that wide (largest rack was 54").
In any event, we trimmed off everything, All muscle, eyes, tongue, skin, ears, etc. We wished we had brought some 20 Mule Team Borax because we could not buy it in Bethel. Then we wrapped the cleaned skulls in some old t-shirts and bound them with cordage. Next, we nested the two racks and protected the tips with lots of foam pipe insulation from the hardware store, using lots of tape. Then we rapped wrapped them all with two tarps and lots of duct tape and lots more cordage. The entire bundle weighed only about 140# and was flown as FREIGHT, not as baggage. They arrived in Kansas City about a week or 10 days after we arrived at home. One of us had to drive to KC and fetch them. The key was shipping as freight, AND having a known freight agent/shipper, which we had. Apparently, not just anyone can ship freight by air.
It was not cheap.