Steven, you obviously have no idea about the legal basis. If you kill or injure a neighbor’s animal unprofessionally, it is in any case property damage and animal cruelty. This does not fall under necessity, and certainly not under [Nothilfe] or [Notwehr]. Cf. §34 StGB.
So the OP will have to find another means, as if it were a wild animal. So some kind of technical devices, scents, or just the water hose. Poison, traps, or weapons are not allowed.
Maybe the neighbor’s cat from andimann is also disabled, who knows. Normally cats cannot be trained to that extent.
Personally, I have no problem with cats in urban and rural environments. With dogs, rather more. But as soon as I see a cat hunting in the fields, there is trouble. There is no acute health risk caused by cat feces either. Worm infestations in children are more of a hygienic problem caused by the parents who do not teach hand washing. The eggs are picked up from all sorts of surfaces, but especially from fruits and vegetables.