A few years ago I got multiple cease and desist emails from a FAANG for a Chrome extension they didn't like. I ignored them and they eventually went away. Sending scary sounding emails to indie devs is extremely low cost.
How could the OP make it a higher cost without necessarily having to get a lawyer themselves? This kind of behavior by LinkedIn leaves a bad taste in my mouth because it’s essentially bullying and I wish there was a way to bleed them a little by forcing them to have their $500/hr lawyer respond to some inane but legally necessary requests.