Bryan - Should you choose to live trap, please don't drop them off at your local farm - I have a dairy farmer friend that people are constantly dropping off cats at. He doesn't need more cats anymore than anyone else. I also don't mean to preach, but if you did use bait out in the open you can catch other things like hawks and owls - they don't generally fare well in a leg-hold trap. Finally, if trouble came knocking on your door and it "got out" to the press there would likely be another wave of anti-trappers coming out of the woodwork. (Here's a good example of how things can go wrong - a friend of mine many years ago had the same problem as you, but he lived in a very urban area. He was a great shot with his blowgun and pegged this problem cat in the lungs with a dart. It ran under a bush on his front lawn. He had to get something more effective, so he grabbed his bow. The cat moved as he shot and he lodged the arrow into it's skull but didn't kill it. It ran across the street with an arrow in it's head and through the neighbor's lawns, never to be seen again. My friend was scared for a week that the cops would find him.)
As for the laws - well, here goes. If a tree falls in the woods but no one is there to hear it, does it make a sound? However, if you get caught killing a cat you will certainly have some explaining to do - maybe a little, maybe a lot depending on the agency investigating. Counting "against" you is the first law, your "defense" is pretty clear in the second law quoted. I've added some bold key points.
NYS Agriculture and Markets law covers domestic animals. It states:
§ 353-a. Aggravated cruelty to animals
1. A person is guilty of aggravated cruelty to animals when, with no justifiable purpose, he or she intentionally kills or intentionally causes serious physical injury to a companion animal with aggravated cruelty. For purposes of this section, "aggravated cruelty" shall mean conduct which: (i) is intended to cause extreme physical pain; or (ii) is done or carried out in an especially depraved or sadistic manner.
2. Nothing contained in this section shall be construed to prohibit or interfere in any way with anyone lawfully engaged in hunting, trapping, or fishing, as provided in article eleven of the environmental conservation law, the dispatch of rabid or diseased animals, as provided in article twenty-one of the public health law, or the dispatch of animals posing a threat to human safety or other animals, where such action is otherwise legally authorized, or any properly conducted scientific tests, experiments, or investigations involving the use of living animals, performed or conducted in laboratories or institutions approved for such purposes by the commissioner of health pursuant to section three hundred fifty-three of this article.
3. Aggravated cruelty to animals is a felony. A defendant convicted of this offense shall be sentenced pursuant to paragraph (b) of subdivision one of section 55.10 of the penal law provided, however, that any term of imprisonment imposed for violation of this section shall be a definite sentence, which may not exceed two years.
New York State Environmental Conservation Law covers wild animals. It states:
§ 11-0529. Cats hunting birds; dogs pursuing deer or killing other wildlife in certain areas.
1. Any person over the age of twenty-one years possessing a hunting license may, and environmental conservation officers and peace officers, acting pursuant to their special duties, or police officers shall humanely destroy cats at large found hunting or killing any protected wild bird or with a dead bird of any protected species in its possession.
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5. No action for damages shall lie against any person for the killing of a cat,...as provided in this section.
So, tell us, what kind of birds have you seen this cat killing?