Thanks for your note - details for the Adventure Cycling North Star tour are found on this web page: http://www.adventurecycling.org/guided-tours/self-contained-tours/2014-north-star-i/ We are taking 60 days to make our 2,725 mile journey.
if you go to the itinerary tab, you'll be able to suss out the roads we are using. To make it a little easier, here is the general routing:
Missoula -> Whitefish -> Eureka -> Icefields Parkway (not over Going to the Sun, but the west side of Glacier National Park)
Alaska Highway (just a connector piece, about 20 miles on this highway)
Robert Campbell Highway (40% paved, 60% gravel, or – 217 miles gravel)
Top of the World Highway (0% paved – 79 miles gravel)
Taylor Highway (60 miles paved, 36 miles gravel)
There is approximately 332 miles of gravel on this set of roads (indicated above), which is mostly hardpacked surface. I would think your 32mm wide tires would be fine, but if it were my bike, I'd put as wide a tire on as I could fit - I know that 38mm wide tires fit on a LHT, and wider may fit too. The wider the better on gravel, in my experience.
I am riding this route this summer, and will be doing it on a bike with 50mm Marathon Supreme tires and panniers.