The FBI doesn't tell the prosecutor what to do. If the FBI finds probable cause a crime was committed, they forward a report to the prosecutor. If the FBI does not find probable cause a crime was committed, it does not forward the results of its investigation. I would guess that most FBI files are never sent to the DOJ.
If the FBI forwards a file to DOJ, that means (without more) that the FBI believes there is probable cause a crime was committed.
That said, I think this case is different. Lynch describes a "team" of both FBI and prosecutors. The same principle applies though. The FBI side of the team provides evidence, the prosecutor side decides whether or not and how to proceed with charging.
-- If either of them fail to do what the law demands, the government will effectively become illegitimate ... --
We're past that point already. Thousands of examples, few of which make the news. The Courts make up stuff to suit social and public policy, Congress long ago abdicated it's role in keeping the executive and courts in check. The government's authority flows from the barrel of a gun.
At this point, that's all they have. They don't have legitimacy and they certainly don't have Constitutional authority for their lawlessness. The compact is broken.