I've taken quite a few with a .36" revolver. Anything smooth bore would work good, .50" on up, or perhaps a bit smaller, as far as shooting shot loads go. As long as the grouse stay on the ground, which they often do, (around here anyhow) a shot load would be great, I think. .58 to .60 caliber would work more better of course. But...I tend to think in terms of "economy" of powder and lead. A .36" round ball, and 15 grains of powder would extend the powder and lead supply compared to larger quantities of shot and powder, that something .50 and up would consume.
I don't think I'd ever attempt to hit one on the fly, with a pistol!!! A 58X58 would be sweet, especially with one barrel rifled, and one smooth.