The fingerprints aren't sent to the FBI clearinghouse. They are scanned and held until someone else rents that locker. After you remove your stuff, the locker is then available for someone else to use with thier fingerprint.
You don't have to scan your index finger. You can use any finger or thumbprint, even your child's fingerprint.
You probably need to work through the paranoia as my bank requires a thumbprint scan whenever we cash a check. I think this is getting more and more common.