If other people on that community board knew it was your dog and the dog was safely secured in your backyard? I think you could send him a certified letter asking him to stop harassing you and your dog on social media when you've done nothing wrong.
I had a crazy neighbor who sent me hate mail about my dogs lol. I knew who it was, the same old geezer who tottered to the end of his driveway that overlooked my yard, to glare at and yell at my dogs when they noticed him and started barking at him. He finally stupidly called the cops to report that my dogs were tied up and barking one night..the result being that the cop parked his car about a block down and sneaked up to my house at 10 pm with a flashlight, TERRIFYING my daughter and her girlfriend who was spending the night. My dogs weren't outside, they hadn't been outside, we were all in bed. The idiot geezer heard some other dogs barking. And the dogs he DID hear were two houses down from him.
You live in town. Dogs bark. If you don't like it, #1, get yourself a private estate, or #2, die.
I told the cop I'd gotten hate mail and that the old fool had made a hobby out of tormenting my PIT and my ST. BERNARD to the point that they started slavering the minute they saw his dumb ass. Hate mail and cop calls immediately stopped.