I'd suggest that you work with the NJ BMV as best you can. Be patient and very polite, recalling that (1) mopeds are not common vehicles and thus the BMV employees aren't familiar with them, (2) the BMV employees are required to work within a rather inflexible set of rules, rules that often don't seem logical even to the people who administer them, and (C) a job at the BMV is not usually a pleasant work environment.
What I did here in Ohio was to compose and write a fairly formal e-mail to the 'contact us' address on the BMV web page. The letter explained my particular situation and what I hoped the BMV would do about it. It took several exchanges, but ultimately the Ohio BMV did what I'd asked (which was to register a motorized bicycle I had just built.) Again, be patient and polite.