To me it sounds like a lazy salesperson that doesn't want to take the time to get it weighed with you. They may not have scales too. I would see if it would be possible to hook it up and tow it to the nearest certified scale. That way the liability if any doesn't lie with the dealer. That was their stated issue to begin with so now you both can win.
FWIW the dry weight of the TH is mostly meaningless unless if you're always going to tow with nothing ever in the trailer. For that reason I would just make sure your TV can handle the trailers listed GTWR . If the trailer CCC is small and you think you might be busting the limit then you need to get weights of all your gear and then get weights of the trailer for sure. Add them both together and make sure to account for water and most gear and voila ..