I used to hunt birds in the Snake River breaks west of Pullman in Whitman County a lot back in the late '70s and early '80s. Same problem. We "solved" it different ways. One of my buddies carried a slung Model 70 .270 and kept his Model 12 in his hands. He was one physically tough young man. And I can't swing a shotgun properly when carrying a slung rifle. Chris could.
I carried a .30-30 T-C Contender 10" handgun with a scope. And only got one doe with it in almost ten years of hunting. The only nice buck I got when bird hunting was a snapshot at about 15' with the full choke barrel of an Elsie 16 loaded with plated 5s (impulsive, and highly illegal). Blew a rathole in his neck and head (at least I was leading, for once....). I still have the antlers with a couple of scores on them from fliers....
I still think the handgun/scope/shoulder holster "solution" is the one that most can afford. A drilling would be the best solution, but they are pricey, and if you have old eyes, you may need to scope it, problematic for bird shooting for most of us.
At least in EASTERN Washington, buckshot wouldn't be practical: ranges tend to be too long. Slugs, I don't know well.
I tried to carry my Valmet/Savage 2400 12ga/.308 combo when bird hunting in upstate NYS during the deer season, but gave up on that: didn't like the possibility of launching a .308 at a grouse in a moment of passion....(and the balance of the combo wasn't right for grouse). Went back to the handgun; an unscoped (because the ranges there are short) Ruger Bisley .357 7 1/2". Worked well; took two nice whitetail bucks in 19 seasons, a handful of does, and several coyotes.
