There is no need to treat this like a crime, or for anyone to try and make sure they are not discoverable off-Fiverr. That’s just silly. Anyone who uses the same profile image on Fiverr as they do elsewhere is easily discoverable via a reverse image search.
To be honest, I wouldn’t do this. I would recommend not responding to the buyer. However, I would only raise this with CS if the buyer references sending an email in a message on Fiverr.
CS is notorious for misinterpreting messages. For this reason, binging an off-Fiverr email to their attention could do more harm than good.
There is an infinitesimal chance of te above happening. However, it is not okay to relate content or details of personal messages you receive to other organizations, any more than it is for Fiverr to email your website saying “Hey @graphtersawyer this guy emailed us today.”
Just ignore this email and leave things at that unless your buyer mentions it in your Fiverr inbox. This will also likely see you continue to get orders, as your buyer will simply assume that you never received their message.