Yoicks. It seems to me the question is at least two-fold:
How to stop the cats from pooping in your yard: let's say these were your own cats. Then a large litter box or a patch of dryish dirt or sand underneath some bushes where they could dig to their hearts content would probably keep them happy. Maybe supplemented by some kind of repellent on the grass, but I don't have any experience with that. Cats are creatures of habit, so you need to break the habit.
How to stop them coming into your yard: honestly, I would go for spraying them with water. Or a light newspaper swat over the head, makes a lot of noise but doesn't hurt. That's what we've asked our neighbours to do when our nosy, bold cat comes visiting unwantedly - yell, shoo, hiss, spray them, be obnoxious and scary, just don't hurt them physically of course.
How to get on with your neighbours...: now that's the hard one. Can you pretend to be allergic? Scared of cats? Have a litter box for a while, then give it to your neighbours as a "gift"? Camouflaged as a really big plant?
I don't know. I'm surprised the cats are that bold, actually, when you have a dog. Can you have your dog chained outside long enough to give them the idea that your patch is the dog's territory?
Winter riding is much less about badassery and much more about bundle-uppery. - malkin
1995 Kona Cinder Cone commuterFrankenbike/Selle Italia SLR Lady Gel Flow
2008 white Nakamura Summit Custom mtb/Terry Falcon X
2000 Schwinn Fastback Comp road bike/Specialized Jett