Some years ago I was renting a townhouse (one in the middle of six units) and they started having a problem with rats. Someone noticed the kids in the complex throwing rocks at the dumpster (they got a kick out of watching the rats scurry out the drain hole when the rock hit the metal) and complained to the office about it. The office called the trash company and they brought out a new dumpster. The office did nothing else though, so those rats started venturing out to find food....
We had a bird feeder hanging on a length of lockwire in the tree outside just beyond the patio. Every night one or two rats would slink out on the branch and down the wire. I had a pump-up air pistol- more velocity than the one in the OP- so I would give it 12 pumps and ease the slider open. Many times they'd be gone in a flash from the noise, but sometimes they'd just climb back up the wire and wait on the branch.
I killed probably 40 rats over the several years I lived there, and learned a few things about shooting rats in the dark.
If you shoot at the rat and miss, you'll know because it will run away. If you hit it and wound it, it will jump straight up (like it got a bee sting), then run away. Shooting it such that it dies instantly and falls on the ground is rare.
Some memorable shots?
- Hit one and it ran away, and the other rats devoured it in the night.
- Hit one of three out along the rock wall, It was a family apparently and I'd killed the mom. The dad ran away and the baby cried from up in the rocks- wife wasn't happy.
- Hit one square between the eyes and it fell dead out of the tree.
- Neighbor was too lazy to carry their trash bags 60 yards to the dumpster and was collecting the bags on their patio

which attracted a rat. It was running back and forth between the rock wall and their patio, pretty much along the same path. So I simply aimed at the ground and waited until he ran under my sights and popped him. He scurried under the trash bags screaming all the way. I left a note on the neighbor's door telling them I'd shot a rat and it was hiding under their trash bags. They immediately cleaned up their patio...
Air pistols are very useful for rodent control when other options aren't allowed or practical.