Probably flying with Bushwacker air.
I've hunted sheep in the Eastern Brooks range since 2011 and caribou are way usually out of the mountains by August 7-8 when we get there. They have already migrated out to the flats by then, although there are a few stragglers, they are few and many miles in between.
So hopefully your pilot can put you on them, whether they're in the mountains or the tundra.
I personally would not fly with Bushwhacker. I think it's a racket the way he's set up. You will pay him $2500 just for a flyout from Kavik. Talked to a guy last season that used him and it cost him I think about $700 or so to get from Deadhorse to Kavik. Plus who knows how much to get to Deadhorse. Plus, that's in a supercub!
You can get flights from Fairbanks with Wrights for $2300/person, to the same spots. Plus you're going in a helio whith a pay load of 750lbs. Now this is for sheep hunts and I've flown with one partner and come out with all our gear, two sheep, and two caribou, in one trip.
I've flown with Wrights every time I've been into the Brooks and they are the best, hands down. Bushwhacker is a one man show and when he has plane troubles, which is known to happen, you can expect long wait times getting in or out. With Wrights, they have a bunch of planes and awesome pilots and are always on time (weather permitting) and have always been able to get me out early.
I'd suggest calling one of the bios up there about the Porcupine herd though, they are not the same as what guys are hunting out of happy valley.
Do you know where you're going yet? Since you're gonna be in deadhorse, I'm guessing you're going with bushwhacker? Id suggest 70 north in happy valley and almost anyone else before bushwhacker. 70 north flies Helios too so you'll have a 750 lb payload vs 50 lbs in a cub.