Frankly if the Japanese of this timeline tried to go overland they wouldn't even reach the US before being defeated.
During WW2 the Canadian army enlisted 730,000; the air force 260,000; and the navy 115,000 personnel. Over half of these people never left Canada and they varied wildly in quality and training, but Canada had the manpower to meet a Japanese offensive from Alaska. With a potentially hostile Japan on the border, this timelines Canada would likely see even more soldiers enlisted or conscripted and better training.
One critical disadvantage for Japan is that during WW2 Canada due to its size and location was a major training center for pilots.
Canada was the primary location of the British Commonwealth Air
Training Plan, the largest air force training program in history.
131,553 air force personnel, including 49,808 pilots, were trained at
airbases in Canada from October 1940 to March 1945. More than half
of the BCAT graduates were Canadians who went on to serve with the
RCAF and Royal Air Force (RAF). One out of the six RAF Bomber Command
groups flying in Europe was Canadian.
Much of this training happened in the prairies.
Going through the Yukon and into the Prairies
Now as others have said, the terrain between Alaska and the populated part of Canada is hellish to put it nicely. Even more than Russia, Canada could afford to lose the land, drawing the Japanese further in and away from their supply lines. From Anchorage Alaska to Edmonton, Alberta (the northernmost city in Alberta and essential for a Japanese invasion that doesn't hug the coast), it is 1,911.6 miles using modern roads. That is 36 hours in good weather using cars of today. Back then 99% of the way is wet rocky tundra, thick forests, swamps and rivers. To get around that area people used rivers to get close to where they needed to go, and then built a short, light railway or more likely a small dirt road, or they flew in by bush plane.
In 1940, there were maybe a hundred villages and towns in the Yukon, Northern British Columbia and Alberta, most of these were Indian reservations, work camps, and fishing villages, nothing that would support more than a few hundred people. The Canadian forces on the other hand would have a steady stream of supplies and soldiers coming by rail to Edmonton from across Canada and the US, then they'd be driven north before being put into boats to go up river. The Japanese would find every kilometer they took harder than the last.
Initially garrison forces that know the terrain could set up defense at the numerous rivers that cover the land. Trying to move across ice covered rivers while under fire and in bitterly cold weather, would be bad for the Japanese. December temperatures in the Yukon typically range from -10 to -21 Celsius.
Even once they get into Alberta it doesn't get better. Alberta has the same temperature range as the Yukon, few roads at that time especially up north, and the Japanese would be going through Boreal forests for about 300 or 400 miles before reaching the prairies and civilization. That's thick pine forests, swamps, bogs, and because its winter, ice, hip deep snow and no roads.
Like in the Yukon the defenders would be closer to their supply lines, many would have grown up in terrain like this, and they could retreat far more easily using small paths and roads known to the locals. They would also be close enough for the several thousand pilots training in Canada to attack while the Japanese have zero air force.
If the Japanese somehow reach the nearest major city of Edmonton, they will be fighting in either wet prairies if in the spring, or hot, dusty prairies if it's the summer with no supply lines. Whereas the Allied army would be well supplied by a highway and railway from the south, with another major rail line to the east and west. Having at least several months of knowing exactly where the Japanese are going to, the Canadian and US forces will have the chance to fortify the entire way.
The Yukon to Alberta to the US route would see the Japanese forces whittled away initially by light Canadian forces using dogsleds, cold, and treacherous terrain. After that while at their weakest they'd be hit with a sledgehammer of Canadian and American forces backed up by hundreds if not thousands of planes that control the entire sky.
Going Down the Coast
If the Japanese avoided that and tried to roll up the West Coast, it's still a death trap, but even worse.
The terrain is no better, the thick forests and rivers would slow them down, and instead of swamps they have to go through narrow strips of coastline between the Rockies and the Pacific. This leaves them wide open for the combined US and Canadian fleet (yes Canada at one point had a fleet, it was mostly destroyers and anti-sub ships but it was there and well trained) working out of Seattle and Vancouver to pummel them as they slowly made their way through these natural choke points.
Again there is the air force flying out of Vancouver and the aircraft carriers to dominate the sky. While the Japanese air force 'may' be able to support its own side from Alaska they would have farther to fly so would have less fuel and time to contest the sky.
During all of this, Canadian and American forces could be sent in by boat to fortify and defend the coastal villages, and behind the Japanese lines to cut off whatever supply train they developed. With the American/Canadian control of the sky and sea, the Japanese couldn't stop it. Any stragglers would be captured, attempts to reinforce the Japanese army would be met by aircraft and battleships supporting the Allied soldiers. While the Japanese bash their heads against the defenses in front of them they'd have to defend their rear, without any chance of resupply.
They wouldn't come within 500 kilometers of Vancouver.
A naval invasion into Vancouver or Seattle, would see the Japanese forces fighting the larger joint naval and air force of the Americans and Canadians. If they did manage to land, they are fighting in urban terrain against tens of thousands of Canadian and US forces.
The results would make the Dieppe Raid look like a well thought out plan.