18 USC §912 provides that:
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.
I do not think that in the context of a movie, play or game the person portraying an FBI agent or other LEO is "acting as such", and surely this does not involved obtaining "any money, paper, document, or thing of value".
The Wikipedia article on Police_impersonation says:
Dressing up as a police officer in costume (e.g. for Halloween), or pretending to be a police officer for the entertainment purposes or a harmless prank toward an acquaintance is generally not considered a crime, provided that those involved recognize the imposter is not a real police officer, and the imposter is not trying to deceive those involved into thinking they are. Nevertheless, replica police uniforms sold in the UK must not be identical to the uniforms currently used by the police, and traders have been jailed in the past for selling on genuine uniforms.
Many films and TV shows have portrayed FBI agents, in some cases actual agents by their real names. This goes back as far as the 1935 moviw G-Men starring James Cagney. Many examples are listed in the Wikipedia article Federal Bureau of Investigation portrayal in media, and many of them used realistic badges and depictions. They were not treated as criminal impersonations, even when the FBI or its officials strongly disapproved of particular movies.
The somewhat similar US federal law 10 USC 772 prohibits wearing military uniform by persons who are not authorized, but paragraph (f) provides that it is permitted:
While portraying a member of the Army, Navy, Air Force, or Marine Corps, an actor in a theatrical or motion-picture production may wear the uniform of that armed force if the portrayal does not tend to discredit that armed force.
A Vietnam-War-era court case held the restriction "if the portrayal does not tend to discredit that armed force" to be an unconstitutional restriction of speech, and held that unauthorized wearing of a uniform was only punishable if there was charged a proved an "intent to deceive". I would expect a similar limitation of 18 USC §912.
That said, if the email was such that a reasonable person might well be deceived into believing that it really came for an actual FBI agent or other government employee, there might be a problem. The suggestions on that point in the answer by user hszmv seem reasonable to me. An imaginary "Confederal Department of Interrogation" say, keeps things firmly in an alternate reality.