Frankly the answer is "make the cops do their fucking jobs" not "make a list of gun purchasers public." Public means that you or I could access the list, what the hell do you plan to do with this list? Tell the FBI "hey that list you maintain has a new entry, as you know, because you're the ones keeping the list?" Do you have jurisdiction anywhere on the entire planet? The literal only reason to make it public is to have a handy list of what houses it's safe to break into when occupied vs when unoccupied. It's basically a treasure map to arm criminals.