This is actually a standard go to for me in civil litigation because litigants and attorneys often make this mistake and commit technical blackmail though they don't even realize it.

You can tell the employer something embarrassing, but you can't threaten to do so in exchange for property. "Property" can including rights to do, or to refrain from doing, something. Here, nothing illegal occurred, yet the feds are saying "if you don't refrain from doing this perfectly legal thing, we're going to get you fired from your job." There may also be due process claims, bivens claims, and/or 42 USC 1983 claims against the officers themselves. I note that the officers are infringing legal speech activity for one.

The problem here is something akin to the Streisand effect though. If you fight them, the "secret" will be revealed. This is of course what blackmailers rely upon for success.