Ken has good advice.A cop cannot search your car without a warrant unless incidental to a felony stop and arrest. Reasonable suspicion, called the "Terry law" by some, says he can pat you
down if he has supportable apprehension that you are armed and that may impact his safety. He cannot search your car
under that so called 'law.' Probable cause can get him to stop you, but does not allow for searching your car in and of itself.If he stops you for suspected DWI, for example, and then arrests you for it, he can search your car--again, incidental to an arrest. Federal law does not protect you from transporting a firearm, intrastate, where state law limits your right to do so. And, an illegal search could find you have a RPG, but it would not be admissable in court under the "fruit of the poisoned tree" rulings.
You asked about the rental property, and I believe you are safe in having the gun there. But, unless coming into or leaving the Peoples Democratic Republic of NJ, leave it at the property. If you don't shoot anybody, no one will ever know it was there. So, if you don't break the law, you have nothing to worry about. If you're planning on breaking the law, do it in CT. They have nicer jails, I hear.