It's great to see that you are looking at ways to get all of that wonderful moose meat back home with you.
Numerous empty storage bins stacked inside one another is a common sight on Alaska flights going in and out of the bush communities...so that part of your plan is solid.
Live2huntelk gave you some helpful links.
I've moose hunted out of Bethel for the past seven years (and will again next year), and I get home all of our meat. It's different for me because we use an in-state air-freight company...it goes right in their freezer on the bone in Bethel and stays in the meat bags for transport to ANC...it is so convenient.
I think a major step for you is to plan and prearrange how you are going to get your meat cut down to bin size and be assured of freezer space in Bethel in advance of shipping so it can be frozen before flying home with you. I am pretty sure that Alaska Air requires your meat to be frozen to fly...you should check.
Contact the transporter service that you are using and ask about options and prearrangements to freeze your meat in Bethel before transport home with you...don't assume anything (especially freezer space in Bethel). Most always our meat is picked-up in the field well before we come out, so everything is prearranged for us with our transporter. It would be a super tight window to come out of the field, cut down the meat to bin size and get it well frozen (not likely) before jumping on a Alaska flight home the next day. I'm sure your transporter will help connect those steps for you.