We hunted for 5-6 days on each of the 3 trips to SK over the past 5 years. I only become concerned I would run out on 1 of the trips and only because, one in our party on one of the trips, didn't seem to have any boundaries on the shots he took. I ended up giving him a box of shells and still had enough to finish out the hunt.
I think 200 rounds is sufficient unless it's an extended stay. FYI, At a few of the sporting goods stores we've been in, I haven't seen anything resembling a good stock of 16 ga ammo and since I hunt everything else but waterfowl with a 16 ga, I'm always on the lookout for 16 ga ammo.
Hunting ptarmigan in AK isn't usually as high a volume shoot like a SK waterfowl hunt, so taking a couple of hundred rounds should be more than sufficient, I would think.