Such a short trip. Rent a small RV (if one can be had). This will allow you to travel just about any where, stay in out of the weather,
see as much as you can in different places. Stop when you are tired, sleep along a river or just about any where you can park it. Take your own food. (1 week is way too short, did I state that). But if you rent a RV, you can try and avoid the weather.
If it is raining on the Kenai Penisula then head north. If it is raining in the interior(Fairbanks, Denali) and it is nice toward Tok or Copper river area, then you have options for such a short trip. To me, it give flexibility. If you figure the cost of these RV against the cost of hotel room, (300+ night), and car rental, 1400-2000 for a week. It might just be a wash and give you flexibility.
As for Denali, the road is only open to driving by people in late fall and early spring type of thing. IN the thick of the toursit season, only buses are allowed on the road.
If going north and you are near Talkettna, take a flight see tour of Denali. or for more adventure, take a flight that lands on one of the Glaciers near the park. Flying over those mountains are a gas, and an adventure. one soon not forgotten.
Take a small plane filght out of Anchorage, one of the ones that take off and return on the lake Hood, if you have never taken a flight wiht a plane on floats, that is a trip in is self. Lake Hood is the busiest lake for small plane in the world.
Did I say 1 week is too short
Just a thought. What time of year