In the United States if someone makes a false statement about you, comminicates that statement to a third party, and that statement can or has caused financial harm, you can reocover damages in court. If there are similar laws in your country. it's probably worth sending a demand letter to cover the time wasted on investigation.

What 3rd party was involved here?

I think monotype would argue they only sent the message to the company they were shaking down.

Of course, LinkedIn’s ToS might beg to differ. I wonder if the bar is worded like a big and statement like you said, or if the disclosure to the third party has to be part of the chain of harm or something (and what precedent says).