We did something similar to your trip last year. We "sprinted" to Banff, a 600+ day, a couple 400+ days, a couple 300 + days. Spent 2 days in Banff then drove < 300 miles a day, everyday, all the way to Talkeetna.
Because of the fires in the Kenai, we changed plans. Stayed an extra day in Talkeetna, 4 days in Anchorage, and 4 days in Valdez. Then drove mostly <300 miles a day except for a couple around 375 till Jasper. One extra day in Jasper. Then drove daily anywhere from 300 to 525 till we were home.
The days in Canada we averaged about 30 mph for the day. We stopped and saw stuff, took pictures, and had relaxing driving days. But we didn't see all we wanted, you never will.
We made it back in 44 days. I was exhausted from driving, the campers tongue was dragging the ground but it was an awesome trip! I wouldn't do it again that fast ever!!
We did it in a 39' DP pulling a Ford Taurus flat. Not sure of the towing capacity of any class C. Our Class A can pull up to 10k lbs but some can only pull 5k lbs. If you try to pull too much you could damage all kinds of stuff. The Alaska highway is rough on your equipment!
Said all that to say this, if you need to pull a big pickup your upgrade will probably need to be a Class A capable of pulling it.
Your schedule it doable just exhausting!