In the ptarmigan invade Fairbanks post I eluded to using a suppressed .22LR pistol for ptarmigan hunting. Here are a couple of nice male rock ptarmigan I killed last month with a suppressed Browning Buckmark. In this pistol I use CCI 40 grain solid Minimags. Accuracy is good and there is little to no meat damage. I picked up a hen right after these two males with a solid head shot at 20 yards. I use a suppressed heavy barrel 10/22 with 2x Aimpoint (2 MOA dot) and that is a good arrangement for longer shots. CCI standard velocity is the preferred ammo for the rifle.
I prefer to hunt ptarmigan with a shotgun because I enjoy the challenge of wingshooting, but the .22s are fun to take out from time to time. I've used a bolt action CZ452 American with Leupold 3x9 and felt more like I was assassinating ptarmigan than hunting them.
The beagles are okay for bird hunting. They will occasionally flush a bird that I don't see and are good for chasing down the rare running, wounded bird. Usually I take the beagles just to get them a few hours of running. Often they are more trouble than they are worth on a ptarmigan hunt.
I've been out for ptarmigan three times this year and am hoping to go again tomorrow!