Just off the top of my head, an offense of impersonation will have an element along the lines of "doing [a] thing[s] such that a reasonable person [does/would] believe you're a real cop", which [optimistically] would not be satisfied as there would be no actual person being led to believe anything, or the court would [optimistically] not find that its model of a reasonable person would be genuinely convinced by someone on the internet typing "I'm an FBI agent" or whatever.

I bet it could be some interesting caselaw actually, if it resulted in circuit court judges (or whoever) writing opinions about the essence of impersonation, fraud, etc. and what kind of actual or hypothetical agent is needed to make the crime a thing that could have happened. E.G., basically, if you sit alone in a room where nobody else can see or hear you, and you put on a realistic local police uniform and declare to the room that you're a licensed police/peace officer, is a crime being committed (i.e., is the nature of the crime "pretending/claiming to be a cop" or "making an actual person really believe it" or something else)

(could also be an intent element to satisfy, not sure)