When it all gets down to where the hammer hits the primer, it doesn't really matter what the action is, High/low wall, CPA, hepburn, Stevens, pick what you like as far as looks and feels go. A good trigger pull is pretty much a must. The barrel is where all the good/bad things happen.
I have a martini that seems to have a very good barrel on it, the trigger is horrible but with the right ammo it will get things done out to 200. Its very important to test your ammo at 200. While things may look pretty good out to 150, those last 50 yards  can be a real game changer. I have shot groups at about 150 or so that I thought were pretty good, move out that last 50 and the wheels just fall right off. 
The barrel and the ammo is where its at, what action you pics is secondary, as long as its consistent.
Sam

I am not sure of the weight limits for the .22 game, but would think that they would be the same as for the center fire guns. If you put a 3.5 or 4 wiight barrel on a .22, that is a pretty stiff barrel. 
I have a 30" 3.5 taper barrel on my 44-77 Pedrosoli rolling block action. The rifle weights 10.5 pounds, reduce the bore to .22 and that should add 1/4 to 1/2 pounds to the weight. 
Having been the RO for a number of these .22 matches,  I feel pretty comfortable saying that most of the shooters will use rifles that are the same configuration and weight as their big bore rifles that they use in the BPCR matches.  Yes the chickens are only at 50 yards but they are pretty small and just as hard to hit as the full sized ones. Big difference is you have to chase them a bit farther when they are waked by a .22.
This game is one that everyone can shoot, come to a match, talk to the shooters, look over what they use. May someone will let you try a few shots, especially at the chickens. You might end up hooked. 
Like I said earlier. Pick the action you like. put a good  barrel on it, a proper match chamber and good sights and give it a try. Everyone should have at least one very good .22. If its good enough there a number of other games out there you can use is for.  Just because it has a hammer sticking out doesn't make it a bad gun for other games. Just makes it unique is all.  And wouldn't it be fun to shoot against all those fancy door latch rifles and win?
Sam