That would be the case if you fail to display the proper submissive attitude and/or fail to give enough respect and/or props to the people's representative and somehow the nose of a bullet in your possession ends up having a cross cut into it, making it a dum-dum.
Dum Dum means just about any effective expanding defensive round according to a Sgt that lived next to me in Brigantine N.J.. That was back when I had hair but when I asked my neighbor about it he said dum dums were bullets that were designed to explode and tear people up, or cop killer bullets that would go through body armor because of the hollow points.
My point being that what the law says and how the rank and file interpret it may vary.
If'n it was me and I had to do what you have to do - get rid of effective ammo before entering the democratic free people's republic.
Carry LRN in .38.
Possibly have copy of statutes allowing you to be armed and definitely have all credentials in order.
Try really hard to avoid contact with LE in Jersey that is not directly related to your assignment.
Stop on when you get back to Free America and purchase small box of defensive ammo.
Pray throughout.
I did know some really upstanding Officers in Jersey that took their oath and people's rights seriously. My opinion is that they are not the majority. Many of those folks think they are not doing their jobs if they aren't rabidly anti RKBA.
Judges, LEO, politicians, citizens in NJ will authoritatively state that there is no individual right to own guns. Maybe not all of them, but too many people have found out the hard way there are enough of them to ruin your entire afternoon.
YMMV.
Edit - Perhaps this incident will give you some insight. In NJ the blade law specifies a "dangerous knife" A really good Sgt and friend of mine that worked in Atlantic City told me two definitions of dangerous existed :
a.) a knife that could cause a dangerous wound.
b.) a knife being used dangerously
My son, a minority in Brigantine because he was a Christian youth, was stopped one night for suspiciously driving down the street with an attractive young female in the car. His Cub Scout penknife in his pocket resulted in two felony charges.
He lost his seat in Marine Infantry school, was discharged from DEP and had to re-enlist. Community outrage and pressure caused the charges to be dismissed with prejudice.
The officer's LT (the guy who was sgt when he lived next to me), the boy's pastor, his employer, his mother's employer, and a slew of other people protested. The PD knew they were ruining his enlistment by not dropping the charges before trial. They took the liability reducing path of just not showing up for court.
If that story is true, and I swear it is and will happily put you in touch with the Corporal, how far do you want to push your luck in NJ?