Years ago it was much easier to transport a handgun to Alaska via Canada; those days are unfortunately gone. The only Americans that I personally know who have been able to get an ATT are those who actually possess the Restricted Possession and Acquisition License (PAL). I think that you have the option of sending the pistol by Common Carrier to an Alaskan FFL and picking up your own pistol there. Unless you were moving there, the process for either the license or shipping it would be extremely time consuming and expensive.
BTW, the fact you actually own a pistol will give them fits. Most Border Service personnel may now be armed, but God forbid they actually take their firearm home at the end of their shift. A good friend of mine has a PAL for his pistol and an ATT to go to the range and when he declares the pistol upon entering Canada, the border guards look so sad that they can't confiscate it.