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I DID grow up on a farm, and continue to live on one now, and I wouldnt shoot it either. My father was an animal loving cattle farmer, and would NEVER have shot something he couldnt humanely run off or protected against. Cats, and other small verment were caught in traps and released far away form the animals, and coyotes and porcupines were scared off by the four great pyr. dogs he kept. The only time we shot something was when it was suffering.
I agree with trapping it. Though, I would lock your little dogs up until you catch your cat.