I would probably sue them. I wouldn't expect the police to try to figure out where the truth of a private spat lies, even if it's rally nasty and one of the parties actually is actually acting like an asshole. Cops aren't actually that good at resolving complicated factual claims & counterclaims. They especially don't like it.

Open the door to police investigation of such disputes and you will have cause to regret it. You think the "trolls" you are so concerned about are above faking such a harassment case and filing the same charges against you? Are you really confident the cops are going to use their inestimable judgment and come to the right conclusion?