the federal reserve bank is not under, nor part of, the federal government
Yes it is. It was created by Act of Congress, it's board is appointed by the President with confirmation by the Senate. It is part of the US Government.
It's a "Quango" (Quasi governmental organization). It functions as a "bank" "owned" by other banks, but that is a legal fiction. It could be dissolved by Congress tomorrow and it's "owners" could not stop it.
It's part of the government.
It's just not formally part of the executive branch.