I talked to the worthless dog catcher here and sent him this ordinance. He called me back and left a message that said. This ordinance was like the jay walking law. It's not enforceable anymore. Wtf. I even went to the police station to complain gracefully. They don't give a shit.
LOL, that's BS. It isn't like this is some ancient law, like the one that says you have to have an escort walk ahead of your car swinging a lantern when you drive through town. The Natchez municipal code (of which this is a part) was adopted in July 1997. Your dipshit dog catcher and local PD are trying to say that the law falls under the legal concept of desuetude, and that's not for him to decide.
You have one other potential solution that doesn't involve putting yourself in legal jeopardy by leaving poison around the property or letting rounds fly in town. In my home state, people had the right to file private criminal complaints through the local judiciary. Where I'm from they call this a District Justice; other places call them District Magistrates, or Justices of the Peace. This isn't a lawsuit, or a small claims action; it's a criminal complaint. A plaintiff would file an affidavit to support the charges, the magistrate would review it to make sure it met the legal requirements, and if it did, the magistrate would issue a summons to the defendant to appear in court. The plaintiff had to pay a fee to file the charges. The charges were the same ones that would be filed by the police or the dog law officer. You'd file a complaint based on a violation of this particular section of the Natchez Municpal Ordinance. The plaintiff could have a lawyer at the hearing to represent him, or he could represent himself. The defendant could also have a lawyer. Hearings at this level, in my experience, are tolerant of plaintiffs representing themselves and that the plaintiffs don't have a lot of knowledge of how a courtroom works.
Go talk to the District Justice in your county and see what the rules are about private criminal complaints. If you can do it, file the complaint. If you get some friction there, call the local District Attorney's office and ask him why a law that was approved in 1997 won't be enforced, and what he (as an elected official) intends to do about it.
If you plan on going this route, make sure you have some evidence on your side to show that all these cats are originating from your neighbor's property. If they leave out food for the cats, take video of the cats eating on their property. Take video of your neighbors interacting with the cats on their property. Leave a camera running all day if you can that shows the cats migrating from your neighbor's property to yours.
I get that it's a really cathartic fantasy to imagine shooting the neighbor's cats, or leaving poison out for them, but I can just about guarantee you that the laws about cruelty to animals
do get enforced (people love cats and dogs), and if you get caught doing it, the one getting charged with something will be you. These aren't raccoons, or rats, or squirrels, which ordinarily aren't protected by cruelty to animals laws; they're domestic pets. If you're in a residential area with houses nearby, they might even try and gild the lily and add some additional charges like Recklessly Endangering because you were shooting a gun where other people could be hit. Besides, even if you kill a few cats, that won't stop the problem. They've already acquired 45 of them; having a few dead ones just frees up a few slots to add some more, plus then you're going to have a neighbor who might suspect you of being the feline Terminator, and maybe they start watching
your house to try and catch you doing something to the cats. If I'm rich, crazy, and a cat hoarder, maybe I'd set up a nice surveillance system aimed right at your property with 24 hour video surveillance (I could buy a great system for a couple hundred dollars), just hoping for that one moment when the camera captures Fluffy's head exploding while you're holding a rifle. Poisoning them presents an even greater issue, because you have zero control over what happens to poisoned bait once you leave it outside. And once again, poison won't stop the cats from coming into your yard, and a bunch of dead cats won't stop your neighbor from getting more of them.
One thing I didn't see in your post (maybe I missed it) was whether or not you actually went to your neighbor and made the complaint directly to them about their cat hoard. Even if they tell you to go pound sand, it'll help your case if you can tell a magistrate that the neighbors were aware of the problem and refused to address it. And start documenting every occurrence of the cats being in your yard, and any interactions you have with the neighbors about it, because that also supports your case.
Good Luck.