"how do you hit a moving target in a trap without just maiming it?"
When you lower the rifle towards the mongoose, because they can't go anywhere, they skip "flight" and go to "fight" mode. They look up, hiss, and try to attack the business end of the rifle. Really they take "aiming" out of it. Rats on the other hand like to move around in a pattern and pause at the corners. Pick the corner with the longest pause, aim, and wait. They will break their pattern after noticing the addition of the rifle barrel in their environment, but after a few sniffs and a pause will usually return to their pattern. Regardless, a .22 is going to be quickly lethal to a tiny animal like that unless you just graze it.
"No break barrel here. Goodbye vermin."
I looked into PCP air rifles, even talking to people who owned them. Fantastic technology and available in calibers large enough to humanely kill deer, mountain goats, etc. Unfortunately, they are prohibitively expensive to purchase and the ammo isn't any cheaper than a standard firearm, which doesn't make sense to me because you are only buying the "bullet" (pellet) and not the entire round or cartridge. I guess it has something to do with the extreme quality control. The specialized equipment required to charge the air canisters is another downer. Lastly, the seals are prone to failure so you're stuck sending the rifle in for service every year or two. I've even seen models for sale that specifically advertise not only do they have no warranty, they won't repair them at cost either.
Until (if) the technology improves the only real advantage to using them is that they are legal where a comparable firearm isn't, they are quiet, which oddly enough requires an expensive ATF tax stamp, and an expensive suppressor, AND to live in a state where it is legal to own/use one for a regular firearm. If you add all of those prices and inconveniences up, then a PCP rifle's useful niche is as a substitute weapon where firearms are illegal or too heavily regulated to be practical, and you get the bonus of a built-in suppressor.
Not surprisingly the technology for high-caliber, high velocity air rifles was born out of necessity in Europe as a means to circumvent laws against traditional firearms.
This is the break barrel I bought about 5 years ago. It still works well even being mostly neglected. We primarily use it to humanely euthanize meat rabbits but of course it gets used for varmints too. There are probably better models that have come out in the last 5 years, but I don't really know. This one met the quality and price point I was looking for at the time:
https://www.amazon.com/gp/product/B004V9...tl?ie=UTF8&psc=1&linkCode=ll1&tag=pw016-20&linkId=1cae84ea7d171cad14f9ae3a9872f286&language=en_US
Here's a good pellet for close-range euthanasia. They work better for close range than hollow point in the above rifle but are less accurate over distance:
https://www.amazon.com/gp/product/B002N4...tl?ie=UTF8&psc=1&linkCode=ll1&tag=pw016-20&linkId=b09a51d45a45855b5d7d0125771fd7f1&language=en_US
ETA: the part about rats, link
