Considering the timing of your visit, I would recommend taking the Alaska Railroad between Anchorage and Fairbanks. In the winter it runs on weekends only, north to Fairbanks on Saturday and south to Anchorage on Sunday. Since you're on an award ticket it should be fairly easy to fly into ANC and out of FAI, or vice versa. It's a great way to see a lot of the state with minimal effort.
If you time it right, you can be in Fairbanks on the first Saturday of February to catch the start of the Yukon Quest, then take the train south the next day. It would be a nice trip and your kiddo would probably enjoy seeing all the mushers and sled dogs. Definitely a slice of real Alaska.