I have shot any number of flying birds, running game and even a bit of trap with a combination and with a drilling fitted with a scope, in each case used on 1.5x. Both eyes open, focus on the target, swing, mount, shoot, just as if it had no sights.
The Germans seem to make this difficult by mounting big high-magnification scopes up high, often with see-through mounts. This just doesn't work if you want the gun to be ready at a moment's notice for whatever comes along. Instead I prefer a 1.5-6x, mounted nice and low.
To answer the OP, I presently have a little Krieghoff in 16/16/7x57R, with a 1.5-6x42 scope in claw mounts. It is Waldschutz model on a Dural frame, and with 22" barrels and weighing right on 6lb bare (about 7 scoped) it is absolutely ideal for the use you described.
