Hello
Running to same situation in NY with a trailer made by International but on NYS DMV website, registering a homemade trailer is easy. In NY if you have registered anything it would show up in DMV computer however after three years of any inactivity it purges from their system. I was told this by the NYS DMV, so if you dont pay the registration renewal every couple years or dont get the equipment inspected then one can see why NYS DMV wouldnt keep tracking it

So in short I would just register it homemade. Thats what I am doing to make it easier if I ever sell it, just to have paperwork. The person I bought the 17 foot trailer from was an older gentlemen who couldnt remeber where he put any paperwork. He gave me a bill of sale and I called local police department in his town to make sure no one stole a trailer and they said nope, after I explained the situation the police said your all set and good luck!!!