ArimoDave summed it up well for any lasting uses. Accurate air rifles (and pistols) are way under-appreciated by powder burners. I started down the rabbit hole about six or seven years ago, starting with a scoped .22 cal FX Royale 400, 6k psi N2 fill tank, and a 72 ci fill bottle. I then followed up by building a custom PRod, dual air tubes, supplemental LDC, M4 stock, and tuning. My buddy kept the PRod for his uses. Those "uses" have been pounds of Northen Curly Tail lizzards, and Green iguanas in S. Florida. The previous two PCPs are really quiet, and pellet stacking/touching out to an easy forty yards - good enough to nail baby lizaard head shots about 90% of the time (100% within thirty yards.)
Earlier this year, I turned in my N2 lease and bought a 4.5k psi dive compressor. Last month, I added a HW44 pistol, which is as accurate as the FX, and just took a forty-three yard Curly Tail from my sister's second floor. The CT was about three inches long, a girth less than a sharpie, and a head smaller than a dime. I could ramble on and on; I routinely state, "... best money I ever spent." Attached on every SMS corpse, lol.
Anyway, if your rats are delirious enough for walk up head shots, you could do just fine with any Walmart BB gun. If you are going to be taking shots that might skip (into the fence,) then consider the Marauder for the cheapest entry - just like ArimoDave suggested. PCPs are not cheap toys, but they excel at what they do.