I don't know about NJ but in NY you don't have to go through all sorts of nonsense in order to get proof of ownership. I have paperwork for my 350x and I never got a title. You just have to fill out the appropriate paper work, pay the taxes, and DMV will give you a "non-transferable registration". After a year, if you renew your reg you will get a transferable registration. I guess that gives them enough time to do a check to see if the VIN comes up hot? I don't know how NJ works it though. If you have a friend in NY that you can trust you could "sell" it to him for $100, have him register it, and then a year later you would have legal paperwork for it in his name. At that point you could transfer it back to your name. It sounds like a lot of work but its probably easier than taking apart a whole trike just to put it back together on another frame.
You could also register it in PA where they don't ask for a title. You just pay for the reg fees. I know ChrisD has done this with two of his machines. Once you have it reg'd in PA you can ride it in NY or NJ with the plates and papers that PA provides you with.
John
1985 Eddie Sanders ATC330r
1986 Monstecate
1984 ATC70 (awaiting 125cc transplant)
1984 ATC125M
1986 GSXR 750 - Keepin' it old school!
1999 Arctic Cat 500 4x4 plow machine
1999 Arctic Cat Thundercat 1000 (lake lizard)
1996 Arctic Cat Jag 440 L/C (ditch banger)