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It seems like you could recruit small clumsy children in the process of making your yard inhospitable to cats. (If they came to like the cat, would that change your feelings about the poo? Dubious, I suppose.)
If involving the kids isn't an option, I'd recommend running after it yelling when it appears anywhere near you (and the neighbors aren't looking). Animals know when they're not welcome. My housemate's dog goes nowhere near my prematurely cantankerous boyfriend, who made a point of yelling, stomping, and waving his arms decisively when the dog came near.
Get a have-a-heart trap. Catch the cat. Take it to your local animal shelter. Boom. You're done. For god's sake, don't get a damn dog. The chances are too good the dog will hurt the cat, even without meaning to. And for god's sake, don't get a cat. What you will end up with is a $400 vet bill when your cat gets bit, and the bite turns into an abscess. Just go with the Have-a-Heart. It's the easiest solution. Word of advice: do NOT leave the trap unattended. Only set it when you are home and can keep an eye on it. When you catch the kitty, put a towel or cloth over the trap. This will help keep the cat calm. Then take it to the shelter. The end.
If you kill your neighbors' cat and they are able to prove it, you could be charged, fined, maybe even jailed. Hello, you can't go around killing your neighbors' cat and not be held accountable. Grow up, this is the real world, and we're still clinging to some semblance of civil society.
There can't be that much poop, so grow up, sweep it away and go on with your day. There are leaves, branches, sticks, and other real objects of outdoor life staining your pristine lawn too; just rake them away and go on. None of it will harm your children.
There's a sneaky element of untruth in the letter. LW says when he is away on business, there's no one to sweep away the poop, so the kids have to stay inside or play in the poop. If they're old enough to make that decision and let themselves out, they're old enough to stay away from the poop. If they're not old enough, whoever is deciding if they can go out or not and letting them out can rake away the poop and say: "ow, that's cat poop, don't touch it" and be done with it. He said there was no one to do this, so how are the kids getting in the backyard? By themselves? Then they're old enough to know to stay away from teeny tiny cat poop. I bet his wife won't get all controlling about it the way he wants and that's why he says there's no one to do it.
There is no problem except LW is sadistic and is searching for an excuse to kill his neighbors' pet. Wonder what horrible insult they imparted on him to deserve this.
Get some counseling to find out why you're grasping for reasons to kill a neighbor's pet, deal with your anger issues, and learn to deal with them in mature, kind ways. Don't pass this legacy of sadism on to your kids.
My husband and I have had this problem with neighborhood cats. This might seem a strange fix, but so far it has been very successful for us. The best thing for discouraging cats is the smell of a larger, adult male animal about the premises. So, periodically my husband fills a soda can and walks about the edges of our yard distributing this message to the local felines. (One must generally renew after a rain.) It does need to be a man who provides the sample, that's just how nature works.
If sprinkler systems are too expensive, have you considered running a low voltage, live line along your fence? Any local handyman can put one in for you and if you have a privacy fence there's no chance your kids can touch it. If its a chain link, well, then...make sure its nice and low voltage and it will keep them in the yard too.
On the subject of getting a dog for the purpose of chasing this cat away: please don't unless you otherwise really, really want a dog. I know, the farm and all, but here's the deal: dogs are pack animals, and we've made ourselves their pack by domesticating them. If you get a dog as a nightime "catter" only to leave it in the backyard and set up one of those fancy dogloos for it, chances are you'll buy yourself a whole world of problems other than the cat to contend with. If you DO want a dog, get a dog that will have a natural interest in chasing cats, such as a herding breed like a border collie or sheperd. Not all dogs will chase cats.
Keep in mind too, that even if the dog option DOES work, it might work only becuase the dog barks. And barks, and barks and barks. Would that really make life for the kids any better?
That electric fence option is looking like a good one.
Oh, and, really: shame on you for deciding you can sleep at night after potentially killing someone's pet.