Actually I would be amazed if a "well known pest control company" would let an employee use their own personal air rifle for insurance purposes.
BUT. In my experience, for those birds, I prefer a larger (.22 cal), heavy and slower pellet over a smaller faster one. I prefer the thundering whallop over the piercing zinger anyday.
I have many air rifles and BB guns. But it's an old .22 single break .22 pellet rifle that rules the rabbit population around here. It shoots pretty slow that's for sure but delivers a lethal head shot.