Welcome!

By fastest route, you are looking at a 4500+ mile drive which would take 10 days of dedicated driving assuming conditions are good enough to drive the speed limit. My software says this is via Knoxville, Chicago, Fargo, Winnipeg, Edmonton, Dawson Creek, up the Alaska Highway to Tok, then to Anchorage.

It's going to be impossible to predict which route would be "best" at any one time from a weather and road condition standpoint until right at the time of the trip. No matter which route you take, the highway crews are very competent and experienced.

Before you go, you need to order the current edition of "The Milepost" - this is the essential publication to have when you travel to Alaska.

If the conditions are bad enough to require chains, you probably should not be out on the road anyway. However, you may want to talk to a GMC dealer about the feasibility of "cable" chains. Whether you get them or not, I think you should probably buy a set of winter wheels and a set of dedicated snow and ice tires such as the Bridgestone Blizzak or Michelin X-Ice. Put those on and ship your OEM wheels with your household effects. You will be glad you have the winter tires if you are stationed in Alaska.

You can investigate the ferries here. The pet policy is stated very clearly. I have a friend who moved from LA to Fairbanks about 20 years ago, and he took the ferry from Bellingham to Haines. He was driving a rear drive Monte Carlo. He said it was pretty hairy from Haines over the mountains to Haines Junction, but the way they work it when the road is in bad shape is run a convoy.