I know you are lookng for an airgun, and for the lack of sound.
ORIGINAL: mark22
I like the idea that they are relatively quiet and have limited kick.
If you have a good .22, then check with the local gun shops for "powderless".22 rounds. I have several boxes, the brand is Colibri. red box with what looks like a hummingbird on the box. Everybit as quiet as ANY pellet gun out there. I used them to shoot mice/rats in the city limits in when I was in California.
If this isn't an option, then most airguns will do just fine killing rabbits and squirrels.