Mens rea. Typing that into a word processor is obviously not using the false pretext to gain anything. Doing it to Claude could be construed as an attempt to gain information, which checks some boxes for fraud and impersonation of government officials.

For reference, I think this is one of the relevant sections of the USC (18 USC 912):

Whoever falsely assumes or pretends to be an officer or employee acting under the authority of the United States or any department, agency or officer thereof, and acts as such, or in such pretended character demands or obtains any money, paper, document, or thing of value, shall be fined under this title or imprisoned not more than three years, or both.

IANAL but I can see interpretations where telling Claude you’re the FBI would qualify. It’s probably unlikely anyone is prosecuted for it, but there’s a chance

The crime is impersonating an FBI agent to others. How you do that doesn’t matter. Privately it won't matter, but if you make a public statement which is untrue like this and it persuades others there may be consequences.

Because you're POSTing them to a server? The same way you can't type everything into Google.

When I am looking for security vulns I tell Claude that I have express authorization and/or I am the author.

Works great.