I have acquired quite a number of pistols (multi-pump and CO2) plus several pellet rifles. My Crosman 1377 (.177) is capable of killing birds and small pests; but my Crosman 2400KT with a 14" barrel is just awesome! (Okay, it's loud, too). I've killed squirrels with it, shooting in the 600s (IIRC). However, for killing efficiency at a low cost, a lower FPS break barrel (or CO2) is ideal: 625-675 FPS kills small pests for me out to 30 yards (with scopes, otherwise I couldn't see to aim). Though I haven't killed any squirrels with vintage Crosman 2100 or the Daisy 880 (both I got for my grandsons), I do believe that both are capable. The Daisy is just too short for me to aim well; the Crosman is an excellent shooter. I just don't get many short shot opportunities, so my Hammerli 850 or my HW30 gets the bid.
To close, if I could shoot at tree rats only 8-18ft away, any one of several of pistols I would try to use to make the kill. My Beeman P-17 is my favorite finisher pistol for when I've seriously wounded one; and it imparts a kill shot to the head quite effectively. I just wouldn't expect such penetration (only 410 FPS possible) at longer distances. Some 1911s, the Daisy 408, and a few others will produce upper 400s FPS, and that will get the job done at shorter distances. Otherwise, a cheaper Crosman multi-pump (or more expensive one) should do nicely. And, yes, there's even nicer guns to buy....but that's another rabbit whole!
hope you find what you're looking for,
Arch_E