I don't think ANYONE should be able to request a federal background check. I don't think background checks should be allowed for employment or housing. Same as credit checks.
A person is either safe enough to be in society, or not. There should NEVER be an in between state. And before you say anything:
1. Yes I am fine with previously convicted murderers living in the same building as me. Teaching my children. And being my co-worker. I want to be able to trust the "justice" system to allow people back into society when they're able to.
2. Yes I'm fine with sexual predators living in the same building as me. Teaching my children. And being my co-worker. For the same aforementioned reasons.
3. Yes. I'm fine with any person with any crime history that the courts and various other medical institutions have deemed that they can be a normal citizen to do what they want.
For the group of people that "maybe they're safe but we're not sure but we don't want to keep them in prison". Then you need to give them a very good stipend and let them live in relative peace. Also, don't limit their housing. Housing limits should never be done. Think of this situation as a "disability payment", because it is a state induced disability.
What someone did in the past shouldn't matter to anyone if that person has been rehabilitated.