Is there really no way to force Time Machine to do a second, independent backup?
Every backup is a full backup. Your destination has hard links to each file so that if you copy any snapshot to another volume or folder - you have a copy of every file. If you delete a file from one time interval - the other intervals lose no data. As implemented, there is no reason to store duplicate bits on e destination if a hard link suffices to fully document each file being backed up.