In NJ you can only own any of the NFA type weapons (but not a suppressor, they are completely illegal even for Class 3 dealers) by the order of a judge of the Superior Court of New Jersey. The law reads that you must be able to show that the PUBLIC INTEREST is served by you posessing a machine gun.
Good Luck on that.... let us know how it works out for you. To date there has never been a state license issued to anyone.
As far as doing it as a dealer (FFL with Class 3), you need the federal license, and to get that you need to first get the NJ state firearms retail license, which means you need a store, a full time business, and then "maybe" they will issue the NJ license. Maybe not though. All depends on how they feel.
Bottom line, again: Good Luck.
Now, you CAN form a trust in Pennsylvania, let that trust own the gun, store it securely in PA (pay a gunshop to let you keep a vault there that only you have the key to), and visit there to shoot it. You can then take it into other states to shoot there, and enjoy it. Put it back in the safe before you come back to NJ. Perfectly legal.
Or do what I did and move. You'll be glad you did