Background investigations for law enforcement candidates include high school college transcripts, work history, credit history, debt to income ratio, criminal record, state and federal fingerprinting, driver's abstracts, medical and mental health history . They are going to know the you have a firearms ID and whether you have a registered firearms. They type of weapon is irrevelant if it was obtain legally. If they ask why you owe a firearm I would give a simple answer like hunting or sport shooting.