I have just got my ticket for a .22 FAC air rifle for pest control. I have used my Walther RM8 .177 sub 12 with Precision Ballistics slugs for squirrels at short 10-15m distance and it is phenomenally accurate for head shots and always goes through, even body shots through the chest cavity.
However, I won't yet go near problem birds with it in case they get winged and end up off my property. I opted for FAC to ensure I had first time knock-down power with 20-ish grain slugs so there is no risk that they may move off.
It depends on where you are shooting and whether if you only injure something, you can safely take follow-up shots.