Being an FFL would be of no help. While traveling to a range is an exemption to the New Jersey ban on having a handgun in a vehicle, you must be a member of a range to use this and member lists are transmitted to the State Police. A reasonable policeman might let you procede but there is no guarantee. The fact that you have paperwork showing that you are attending a class should exempt you but it would be up to a judge and jury.
Best to stay out of Jersey.