Looking for a small game rifle .22.

A 300 dollar budget is going to obviously keep you out of even the mid-range PCP market once you factor in the cost of the pump and all, but I don’t think you sound like you need a PCP anyway for your small pests. There are some great multi-pumps available that would do the job you want to do easily. The Benjamin 392 is a classic and they are a very reliable air rifle and are very capable in .22 caliber. Accuracy is very decent and scopes can be mounted or you can use iron sights. 
There are also a huge number of break barrel rifles out there that could fill your need, but I’ve been out of the break barrel world for a long time now and can’t give you a recommendation for a 300 dollar model based on my own experience. I’m certain others on here will. 
Check out some videos on the tube of the Benjamin 392. See what you think. 
 
If you mean a powder-burning .22 rifle, I'd say - a Ruger 10/22, I have several (for me and my sons). But, I also have a Ruger American .22 Bolt Action with a Q Erector silencer on it - probably my favorite .22. It's accurate and is as quiet (maybe quieter) than any of my pellet guns with subsonic ammo - and those rounds packs a punch with its heavier than a pellet bullet. It will shoot any LR round I feed it, whereas my auto 10/22's won't cycle sub sonic rounds.