That would require collusion on the part of the FBI. I thought they only had authority based on federal law. Have you ever heard of the FBI arresting people for speeding?
Maybe purchase of a gun that is banned in other states is an ATF matter? I don't know. What I do know is that FedGov has data storage facilities everywhere, racks and racks of info, on everyone, and they're building more all the time. You might think a background check goes only to the FBI, that they process it and then destroy it without sharing with anyone. Or you might think that as soon as your request hits the server, everyone knows what you bought and where you live...