I am a former LEO, and a resident of Alaska. There is a lot of well lets call it strange info in this thread.
I have driven the Alcan at least one way every year for the last 8 years. Every time I have had firearms with me. There are ways to do it perfectly legal. You just need to understand the rules, and do the paperwork.
If you wish to temporarily store a firearm, a safety deposit box is NOT a good idea. Banks actually have rules against storing firearms and even cash. Storing either such item will be a violation of the agreement, and if they find out whats in there. Dont expect it to be there when you get back. The best way to store a firearm when you have no one you trust to leave it with? A firearms dealer. Many if not most firearm dealers will store for a fee, your firearm. Explore some of the dealers in the area where you want to "drop" your hardware and start talking to them.
Another option is to ship it to you. I often send handguns and other "prohibited" hardware to the other side.
I hold CWP's in almost 32 states, and take advantage of that.
When driving North, I also take advantage of cheaper ammunition in the lower 48. And the fact I can legally transport 5,000 rounds thru Canada. I typically remove 1 round from a random box putting me at 4,999rds. Usually they don't care, a few times they have decided I was up to something and count. Then watch the frustration when they try to find the 1 missing round, after I have told them it was left in the US.