Another option to think about is having it processed locally in Kodiak (or even in Anchorage possibly). Many meat processors up here are known shippers w/ Alaska Air Cargo and can ship the frozen processed meat. Not knowing when your hunt is, it can be pretty mild here in Anchorage in the fall and I would worry about my meat sitting in the baggage area waiting for your connecting flights. This is definitely not the cheapest route and I know many find joy in processing their own meat, however the risk involved in losing your meat might out weigh all that. In 2018, while I was living down in Arizona, I came up to do a moose hunt with some old high school friends and was fortunate to be successful. Had I handled the meat as baggage, it would have been with me for an extra day in Fairbanks waiting for my flight, sat in Seattle waiting for connecting flight, to be retrieved at the carousal in Phoenix. I opted to have it processed in AK (highly recommend Delta Meats if you're ever needing processing in that area). They are a known shipper w. AK Air Cargo. Once finished processing, they delivered to airport in Fairbanks and shipped it as cargo to Phoenix. Showed up that night frozen solid.
Just some food for thought. The logistics up her can be quite challenging to say the least.
Good luck on your hunt!