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)

>Because you're POSTing them to a server?

How does that change anything? The HTTP protocol is just how I communicate with the program, just like how the USB protocol is how I communicate with the word processor. The dividing line is when the message crosses computer boundaries? Then it should also be illegal to write "I am an FBI agent" in a text file and upload it to Github.

>The same way you can't type everything into Google.

Who says you can't, physically or legally? Maybe Google will refuse to fulfill some search requests, but that's a different matter from it being illegal.

https://proprivacy.com/tools/ruinmysearchhistory

Here's a site that automatically uses your browser to do questionable searches to get you on a watchlist. Try it! Nothing will happen.