I've used a .22 S&W78g on rats, not a hunting grade pistol and velocity (less than 400 fps) is not really up to the task.
I did find it would kill a large rat cleanly at close range if pointed pellets are used, but wadcutters while producing mortal internal injury would not break the skin and the rat could crawl away to die no matter how many times it was hit.
Those rats stunned to immobility by the first shot would still be kicking after five or more hits.
Basicaly you need a bit more velocity than I was getting from that S&W if you go with .22 bore size.
My Daisy pump up .177 gave quicker kills with either pellets or steel BBs, due to the much higher velocity of around 800 FPS, and much better penetration, often passing completely through the body of the largest rats.