I'd prefer to avoid springers and PCPs and I'd like it reasonably quiet.
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It's not $250 or less but, I have a .25 cal Benjamin Marauder that is a PCP with built in suppression it’s very quiet. Shoots a .25 26 grain pellet @ over 800 fps. I can do less than 1 inch groups @ 50 yards. It will drop all pest animals like a rock probably the most humane way to take them out without using a 22LR. It more fun to see them explode using a 223 but not worth .50 cent a round.