They were tough people there in those days and would have given the Japanese a run for their money. As Guy Gabaldon said, the Japanese were good fighters but they were stupid. Surely they weren't stupid enough to invade Alaska! (The Aleutians excepted) Alaska is six times bigger than the total United Kingdom and inhospitable to say the least.
The Aleutians were always a diversion (that didn't work), but the perceived threat was always higher in the American psyche than the reality. I don't think the Japanese ever intended a major invasion of the US - why enter the hornets nest? But the deployment of troops as to make it look like an attack/invasion was a possibility was a tactical master-stroke as it diverted many resources away from the real front-line. It's just that Japanese (except Yamamoto) never really grasped that even these diverted resources were only a small fraction of America's potential, and so didn't make much of a difference to the overall outcome.
Yamamoto and possibly a few weaker ones knew that if America didn't negotiate a peace in the first year of the war, Japan would lose. Pearl Harbors sneak attach without a declaration only succeeded in raising the ire of the American people. Once in, there was to be nothing less than total war and total surrender expected from the Japanese.
