I've shot a lot of caribou in Alaska. There are several fly in services in Anchorage that will fly you and your gear into the bush, and pick you up at a later date. I'm sure if you take the time to do a bit of homework you can find a fly in service in Anchorage.
Getting a caribou will mostly depend on the weather, as once they start moveing you have to be in the right place, or you just won't see many. Mid september is probably the best time, and the weather could be from cold and snow to warm and bugs. Most septs are cold enough to freeze at night, and fairly warm and decent durning the day. Sept is one of my favorite months in Alaska!
Bugs will depend on the weather, if it freezes at night the bugs aren't so bad, if it stays warm then you will have bugs.
The bird hunting will be for ptaramigian, as they like to stay in the tundra where the caribou are, also you could see a black bear, or moose. Most places where there are caribou, there will also be a brown bear around. I've seen many brown bears and fresh tracks in caribou area's.
Personally, i think it's on the hard side to get both a moose, and a caribou on a short hunt with out moveing your camp. I think it's more likely you'd see a blk bear while hunting caribou, at least that's the way it always was when i hunted caribou a lot there. I've shot most of my caribou SW of Anchorage, and if you go north instead, all of this info will be different.
Drilling Man