Have never lived in NJ, so I have no experience with your DMV, but I have more years of dealing with those folks in California than I like to admit to - maybe a bit of CA DMV experience will be of help.....
On the "how do I tow it with no plate" issue, CA has what they call a "one trip moving permit" for non registered vehicles that need to be moved on public roads - here, you take whatever paperwork you have in to a DMV office, tell them why and where you need to move the vehicle (like to a scale for a weight cert.) and they issue a permit that lets you take it from where it's stored to the destination and back. I'd think NJ probably has something similar.
Are there any states nearby where the DMV is looser on their policies? It's far from the best alternative, but if you're not too far from Delaware, Maryland or PA, they're easier to deal with and you have friends or family there, perhaps you could sell the trailer for $1, they could license the trailer and then "sell" it back to you for $1. Once it's licensed and registered, it shouldn't be a big deal to get NJ registration.