Traps should be inspected daily.
In England it is illegal to release a live rat, they must be killed humanely, drowning in the water butt is not permitted now. An air pistol, pellet not BB (too weak), will humanely dispatch a trapped rat. A Webley Junior can do the deed point blank with ~2.5 ft lb, but about as accurate as a peashooter..
In England it is illegal to release a live squirrel elsewhere away from where it was trapped. They may be killed or released on site.
We do not need a high power airgun to kill a rat. We do need accuracy, rats are small and scurry in an unsporting manner. Being small it will be close range, say out to 20 metres. And anywhere in between, where the faster flatter trajectory of a 177 may be a better choice for rapid change of distance.
An air pistol cannot be brought to bear accurately quickly enough to do the job, an air rifle is needed
Rats are small, a scope will be needed, and that's another can of worms.
My CP2 long rifle, just after I had zeroed the scope, deaded a rat as 20 metres on its first capsule of CO2