It's easy, don't even think about it. Just go. You do NOT need a GS, you'd be fine on your K1200. I'm serious. You can do it on ANY bike. You DO NOT need a giant E&C-approved beheamoth of a "dirt bike".
Alaska isn't hard, especially if you stick to the Alcan. It's just like any other paved two-lane road you'll find anywhere in North America. There's a spot of construction here or there, but it's no problem.
I headed up there a couple years ago with a motorcycle license that was a month old, riding a '97 Yamaha Virago 750. I took that thing to Prudhoe Bay and all kinds of other places that a low-slung cruiser with butter-smooth street tires is never supposed to go.
Seriously. It's not hard.
STOP OVER-THINKING IT AND JUST GO. It'll be amazing!