You go there tell them you have no reg for a trailer this is the exact forms you get handed. I have done numerous trailers and this is what I get handed everytime. Even if you go in with a DTF-802 as you bought it, they dont take it. You still fill out the 803 since your doing it as homebuilt now. I just did a boat trailer the same way. Went in with a sales tax form for what I paid for it and weight slip. Got handed these forms which are the same as I have had before for other trailers. Maybe up to which ever office, but thats not my problem. I give them what they want, walk out with what I want. Simple as that.
Unless the buyer likes paying NYS even more money. They can have the previous owner do the excat same thing, get a reg and a sales tax form. Then sign it over all over. Then pay tax on the sale price and a reg fee. Thats the only other way youll ever get a new reg for trailers. Just as people walk in with out a clue on what they need to register cars. You just need common sense on whats needed. Probably 3/4 of the people waiting in DMV offices are filled with people missng half the crap they need which makes dmv a pain in the ass to go to.