When I came down the Alcan a couple of years ago, I had a very common 15" tire on the Jeep I was towing go bad. I couldn't find another until I got to the States, though everyone offered to order one and have it in 4 - 5 days for me.
I'd take what you have, but take one spare ready to go on a rim and another spare tire with you from one of the ones you changed out as mentioned above.
You'll be able to get 20" tires in Anchorage if not Fairbanks when you get there. If you really run into tire problems, call ahead to a tire shop in Anchorage and have them hold a tire for you. If they don't have it, the ships from Tacoma arrive twice a week and they can have one waiting for you by the time you get there.
Bill
