There are size limits for Bass statewide. There are also seasons statewide for Bass and Snapping turtles. They are not without regulation.
It's also unclear from your original post what fish specifically you were trying to catch at the point you had the run-in with the game officer.
The reality is, there are pretty strict laws in NJ about captive game, and it's not entirely clear if that includes fish as well as mammals and birds. In many states, it's not legal to transport live fish with intent other than to consume them. The rises from the concern that they'll be transported, one way or another, to another watershed.
When it comes to wildlife law, unfortunately it's often up to you to know the law for sure, and most of the time the full laws are not easily accessible, wrapped up in a tidy pdf. waiting to be read. If you're sure you're in the clear, then stand your ground. If he arrests you for breaking a law that doesn't exist, it's on his head.
In any case, sorry these guys are givin' you hell. The DNR officers here in Illinois seem to be stand-up people.
Josh H