Reading this, and LMAO as I went through. Moke has my nod of appreciation, tase the little bastiage.
Actually it requires feeding another mouth, and YES having another crap machine, but if you had a big honking Dog who thought the definition of fun was flipping strange GATO's in the air, and playing rag doll with them, the GATO would cease and desist. As you point out it is a smart critter, and you don't want to kill it outright, but F-ing with it mightily will give you something to watch in the wee hours, and I would think the Gato would look for greener pastures onto which to poop. Unless it is a supreme Masochist, then live traps may be a necessity.
Not too many years ago we had 2 such muttlies, and they loved our Gatos, who just went in a litter box, but God only knows the insults they inflicted on the neighboring Gatos with thoughts of soiling our land. Jake the smaller mini-shepherd, would run them to ground, and Max the huge Lab like creature would just sit on them. If they tried to move he would growl menacingly, and the Gato would calm. Usually he would finally fall asleep, and roll off, then Jake would be on their azz until they finally cleared the county line. Never did they actually harm the Gatos, just f'd with them.
Fun to watch, and no chit show to clean up. Thing was we didn't really need to train them. I think it's inherit to being a dog. My TURF, my Masters don't love you, so neither do I. Vs. Ohhhh yeah, thats one that Master loves. well I can't, well maybe just a little, got to make sure they get their exercise.........