Hammerli 850 magnum, but it is a bolt action CO2 gun. Needs an adapter for extra mod and CO2 Tube so you can use the 12g cartridges. But a very accurate gun up to 30 yards. 22 cal generates about 11-12 foot pounds energy at the muzzle.
I've had an Umarex 850 magnum for about a year and a half now and have to say, that bad boy is one honey of an air rifle! It's a fantastically accurate and incredibly fun indoor target shooter at ten yards, but it also has the power to pulverize any pesky pest with the unmitigated temerity to show up out front or in the back yard every now and then! ;-) I went ahead and bought the 12g cartridge adapter with my initial purchase online at Midway, along with a few extra 'magazines,' all of which were right around a hefty 30 bucks each, if I recall correctly, but I was lucky enough to get the rifle itself for a super affordable, just over one-hundred-and-eighty dollar price tag, back when I (ahem) pulled the trigger on the purchase. Free shipping too, so that was a pretty good day of online shopping! ;-) The adapter is a necessity for me; I really like having the option of using a single 12g CO2 cartridge (well, one full and one empty) when time is short or I need only twenty five shots or so. If I have a bit more time and want more shots, then two full capsules are called for (that second cartridge can be finnicky about being pricked), but when it's maximum shooting pleasure for which I'm shooting (an entire session of target practice or a nice, leisurely day of hunting (when I still could)), then it was time for the 88g cannister, with enough CO2 for right about two hundred plus shots!
All together, I have five very nice CO2 air rifles in my collection, along with a Crosman 2240 pistol in carbine configuration, all of which could be called on for small-game and pests if absolutely necessary, but I use them for ten yard indoor target shooting almost exclusively, when I'm feeling a little gassy. Hyuck, hyuck. My 850 magnum is about the highest quality and most powerful CO2 rifle I have, but I also really enjoy my slightly older, Sheridan-made 2260MB (now only made by Crosman) CO2 rifle, equipped with very well-designed open sights that I absolutely love, as they allow me to shoot the thing with incredible accuracy! Actually, when it comes right down to it, all of my CO2 rifles are highly accurate, including even my somewhat dated, slightly less super-accurate, but extremely fun-to-shoot Crosman 1077W semi-automatic. Also, although it's my newest and so far, least-used CO2 airgun, my super-affordable (just over sixty bucks from someone at Amazon.com a couple of years back) Beeman QB78s is a very, very nice rifle!
I also have a few gas piston air rifles and one or two springer pistols, and the rifles in particular are more than powerful enough for pests and small game, but excepting my quite cool, new-tech-equipped Omnia ZRS from Spanish company Norica, I just don't dig all that weird (to me) movement and kick that comes with a break barrel, and I don't want to even -think- about the PCP side of our hobby, although the type certainly would be up to the job. I mean, suppose I tried one and absolutely LOVED it? Scary thought. Nope, my wife and I need to keep some kind of retired life lid on my airgun hobby spending, and PCP seems like it can get real expensive real quick, so thanks but no thanks. Now, if we were flush with jack and I was healthier and still able to enjoy getting out to hunt in the wide wild world, I might feel differently about the type, but being a bit too gimped up now as I'm approaching my dotage (ha ha, I hope!), I guess I should count myself lucky (yes, that's the way to look at it!) that I don't have the need.
Yep, all of the nearly thirty air rifles and pistols now in my collection are pretty danged sweet shooters in their own unique ways. Still, when it comes to the CO2 and spring/gas piston models I have, I just don't believe they'd be quite as handy and all-around-utilitarian for a situation like that described by the OP, especially when most of my collection is comprised of my overwhelmingly favorite airgun type, the variable pumper air rifle (a number of pump pistols are in there too, of course <grin>), nearly all of which are capable of handling the job, in .22 or .177 caliber. Hmm... So many choices, so little time...
So... I recommend a good variable pumper: preferably one in .22 caliber, such as the Crosman C362 or C2023 (the 100th anniversary version of the 362, now only $300, instead of the $400 price I paid), or the excellent .22 Seneca Dragonfly Mk2, also available in .177 caliber (I have both). Then again, there's nothing wrong with the smallest caliber airguns now available if you prefer, particularly for target shooting (very affordable!). For this situation, I'd suggest perhaps a .177 Winchester 1977XS (excellent accuracy!), or it's little cousin, the Daisy 880 (or the internally identical Daisy 901) would be great choices. More great choices would be the synthetic stocked Legacy 1000 or slightly better (in my opinion) 2100b Classic air rifles, both from Crosman! But wait... there's more: yet another Crosman option would be the roll-your-own Custom Shop at their crosman.com website, where you can design your own variable pump .17 or .22 caliber carbine and equip it with ALL the bells and whistles, for a super reasonable total cost of about two hundred and twenty bucks, delivered! Now that's a great deal, and while I haven't had the need yet to test drive it for the purpose, I have no doubt my own .177 caliber, all-black, super sweet little carbine built from the basic vanilla 1300kt air pistol would make an EXCELLENT tool for eliminating pests at the range and under the circumstances described. Oh, Wait! I forgot to mention you also get to have your very own, personally custom-designed Crosman carbine inscribed on the right side with the name or text of your choice, in one of three fonts you pick -- at no extra charge! I named mine Evie, a nickname for my incredible wife of nearly forty years that had me design and order one as a gift for my last birthday (what a gal!). ;-) I chose to go with a Williams notch (not peeper) rear sight for mine that helps me get fantastic accuracy with almost too much fun, and I love it so much I'm thinking seriously of building myself another, even groovier little carbine in .22 caliber, before prices go through the roof like everything else!
Well, that's my two cents worth.
Good luck, everyone!