Wow, that sucks. Here in California we can hunt a lot of stuff with an Air rifle. It's listed as a method of take under the "general" season for all this:

Chinese spotted doves, ringed turtledoves of the family Columbidae, California quail and varieties thereof, Gambel or desert quail, mountain quail and varieties thereof, blue grouse and varieties thereof, ruffed grouse, sage grouse (sage hens), white-tailed ptarmigan, Hungarian partridges, redlegged partridges, including the chukar and other varieties, ring-necked pheasants and varieties, and wild turkeys of the order Galliformes; and the following game mammals: jackrabbits and varying hares (genus Lepus), cottontail rabbits, brush rabbits, pigmy rabbits (genus Sylvilagus), and tree squirrels (genus Sciurus and Tamiasciurus).

The following nongame birds and mammals may be taken at any time of the year and in any number except as prohibited in Chapter 6: English sparrow, starling, coyote, weasels, skunks, opossum, moles and rodents (excluding tree and flying squirrels, and those listed as furbearers, endangered or threatened species).

Any house cat (Felis domesticus) found within the limits of
any fish and game refuge is a nongame mammal, unless it is in the
residence of its owner or upon the grounds of the owner adjacent to
such residence.

Nongame birds and nongame mammals may be taken in any manner except as follows:

(a) Poison may not be used.

(b) Recorded or electrically amplified bird or mammal calls or sounds or recorded or electrically amplified imitations of bird or mammal calls or sounds may not be used to take any nongame bird or nongame mammal except coyotes, bobcats, American crows and starlings.

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It's great to be a Californian. Though you do need at least a .20 caliber airgun to hunt turkey. What are the laws like in your states?

P.S. live2hunt743, .177 is plenty powerful enough to drop a feral cat at 40 yards if you have 1000 fps and a hollow point.