Another option to think about is just having it processed up here, then the processor can ship Air Cargo. It is definitely not the cheapest option, but neither is hunting up here, LOL! In 2018, before I moved back up here, I came up for a moose hunt and was successful. Had the meat processed...
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...
Do you already have all the gear you need??? If not, is it gear that you would continue to use where you live? If no to either one of those questions, renting might be a viable option. Check out https://www.alaskaoutdoorgearrental.com. This looks like a pretty reasonable option for renting...
Sounds like one heck of an adventure!!! Is it moose hunting if you don't have to fix something though????? Those last pics of the kid digging in are what its all about!
Sounds like you have a lot lined up! Valdez is a small town that gets heavy tourist traffic. That said, they have a tendency to start clearing out Labor Day Weekend. I would suggest talking with the hotel you are booked with to see what their occupancy looks like around the dates you are...