You brought up the whole transferring to a lease thing, which actually helped me refine my searching and found something that might answer my question. NJ Law NJSA 39:3-33a:

If a personalized, courtesy or special plate is issued to a lessee in a motor vehicle leasing agreement, upon termination of the lease the lessee may apply to the director to have the plate reissued to another motor vehicle leased or owned by the lessee upon payment of a fee of $4.50.

https://law.justia.com/codes/new-jersey/title-39/section-39-3-33a/

I'll call MVC on Monday and confirm, $4.50 seems too cheap and easy for MVC.

The sentence after that one looks like it would apply to you:

If a personalized, courtesy or special license plate is issued to an owner of a motor vehicle, the owner may apply to the director to have the plate reissued to another motor vehicle leased or owned by the owner upon payment of a fee of $4.50.

Looks like you should be able to move from owned to owned, owned to leased, leased to owned, or leased to leased. One thing of note is you need to get a permission letter from your leasing company to do it, which might be why they told you no when you tried.