No, there is no attempt here for the FBI to be independent. Yes, it is supposed to be. The same is true for the DOJ.
Now that the FBI has taken over the case and kept the state of Minnesota out of it, it is federal. It was going to be federal anyway, because ICE is involved. But the FBI won't share its "evidence" with any state or local authorities.
Seriously, this stuff is difficult to even write. I know you're used to a much different America, and so are we. But we're in real trouble right now. This is literally an authoritarian situation, where rules, norms and laws are being ignored and the people are being given an alternate reality.