Sweet tag and congrats!
I am supremely jealous. A Chugach sheep tag has been at the top of my wish list for a long, long time.
Lots of good advice here. Best thing you can do to train is walk up hill with a heavy backpack on. The steeper the better.
I've never had a pair of "barefoot" boots or shoes on, but taking a quick look at the model you mentioned, I think trying to pull off a sheep hunt, especially in the Chugach, in those would either 1) really limit the type of terrain to you travel in and manage, 2) create a legitimate safety issue if you get into any technical terrain, or 3) totally wreck your feet.
There is a huge different between moving through this type of terrain "light and fast" and slogging through it for days on end under big loads. Of primary concern, with regard to footwear, are traction, sole stiffness, and ankle support.
Without a half or 3/4 length shank, you won't be able to toe into steeper terrain or plunge step very effectively going down. Without a well defined heel, you won't have much for traction descending steep, slippery, or hard packed terrain. Going down is where most accidents happen, especially if you are coming out heavy.
I'd highly recommend finding a good pair of mountain boots and start getting your feet used to logging miles and elevation gain in them ASAP. I don't think you will find the transition too bad if you are able to find a pair that fit well out of the box.
La Sportiva makes a number of models that have worked well for me with a relatively low volume and average width foot. Scarpa, Crispi, Lowa, Arcteryx, Kayland, Asolo, Garmont, and others all make suitable models. Finding a retailer that carries them in your area will likely be the crux.
As for gaiters, I usually don't pack them on sheep hunts unless I know I am going to be doing a lot of small stream hopping on the approach. You won't notice the strap underfoot if wearing a pair of boots with a heel. You'd probably notice the strap and further reduced traction in a flat-soled footwear.
