I have been using a Chiappa Little Badger with CCI Quiet .22lr as my grouse gun while deer hunting for the last few years. It's accurate enough for grouse. I have it sighted in for about 10 yards. The original sights were terrible. I mounted a super cheap scope from Amazon and that was sufficient, but heavy, so now I'm trying a different sight for this season.
I carry the little badger in my pack with my deer rifle in a Kifaru gunbearer. When I see a grouse, I usually back off a bit, drop my pack, pull out the little gun and slowly move in. I body shoot the grouse. I try to hit at the base of the neck, but sometimes ruin some of the meat. It's only a .22 it doesn't do too much damage.
I tried a pellet pistol, but after shooting two grouse, hearing the pellet hit, then watching the grouse fly away like nothing happened, I decided to get a real gun and stop wounding grouse.
My brother has a Pack-Rifle. Its a slick little gun and only 1 pound. Its not as accurate as the little badger, but it weighs about 1/4 to 1/3 of the little badger and costs about 3 to 4 times as much. If I did it again, I might buy a pack rifle instead of the Chiappa, maybe.