I admit my reference to shipping to an FFL isn't first hand knowledge. That's what friends have had to do, but check with the ATF and they can tell you exactly what to do.
You can always just buy a handgun off Alaskaslist when you get here. There's tons of them for sale private party.
You're probably 100 times more likely to end up in a dangerous encounter with another human, even in remote bush Alaska. People pose a far greater danger than bears. Nailing a moose with your truck should worry you a lot more than bears. Bears are heavily hunted in most of Alaska and the vast majority know humans are dangerous and they will high tail it away from you. You'll probably be really surprised at how few bears you'll see.
Have fun, see the sights, don't worry about the bears.