I don't know about AlaskaAir being the worst. As an Alaska resident, if I want to go ANYWHERE I have to go at least partway on Alaska...I have NEVER been asked to check my mandolins, period. One is in a formed case, the other in an oversize rectangular case (32"x14"x6"). As a precaution, both cases are fitted and have oversized, well-padded headstock nests.

Check with your carrier concerning insurance. I recently sent a mandolin to Seattle on an AK Air flight, and was able to insure it for the full $4500. If they won't insure it, send it as cargo, then you can buy as much insurance as needed.