There are several road trips that are very do-able and extremely scenic, but not well publicized.
1) The Denali Highway between Healy to Paxson is a 3-day, 350 mile route on a gravel road. Incredible scenery, very little traffic, no facilities, camping required. Best with a support vehicle.
2) The Fireweed400 race route between Sheep Mtn (mile 110 on the Glenn Hwy) and Valdez is 200 miles on a paved 2-lane highway with very good shoulders and very low traffic. Again, stupendous scenery including a glacier, raging river, waterfalls, and a big mtn pass. Plenty of roadhouses and one town along the way. Check out the race's website. Their 2-day touring event is supported by aid stations, you really don't need a support vehicle. You can return to Anchorage via ferry to Whittier, pay someone to get you through the tunnel, then bike back to Anchorage.
Biking to Sheep Mtn from Anchorage would be possible except for a 10-mile stretch of shoulder-less highway that, in my opinion, is extremely dangerous (hilly, no sight distance, high traffic volume). If you have vehicle support to transport you over this portion then you have a great trip ahead of you.
3) You can extend the trip to Valdez by continuing on to Cordova by state ferry. The ferry arrives in the evening, so for the first night, stay in a B&B in Cordova (recommended) or camp on the beach (not the nicest). If you are really organized, you can rent the US Forest Service Cabin at mile 20 for the first night. Bike on the Copper River Highway ( a gravel road with almost no traffic) to the very clean and relatively new US Forest Service campground. It is located on the Copper River opposite a glacier that calves throughout the summer - spectacular. You can bike across the old railroad bridge (now pedestrian only) 20 more miles until the railroad right of way disappears into the brush. Return to Anchorage via state ferry.
4) Get on a plane and fly to Nome. From Nome, there are 3 gravel roads emanating outwards into the beyond. Carry as much food as you can on your bikes, camp on the beach or tundra or river banks. Bug repellent required. Return to Nome for re-supply, then do another road. You practically have the back country all to yourself.
5) From Anchorage, bike to Denali National Park along the 2-lane highway with good shoulders and moderate amount of traffic. Camping at pullouts will be required, towns too far apart. In the park, bike as far as you can on the 80 mile gravel road. No supplies in the park.
Of course, all these bike routes come with endless sunlight. However, our weather can be cold and wet. Bear encounters are possible in the more remote areas such as the Denali Hwy, Nome, and Copper River Hwy, especially at the rivers during salmon spawning seasons - bring bear pepper spray and study up on bear avoidance strategies.
I hope these ideas inspire you to come on up !