Is there anyway you can get the dealer to weigh the tongue of the trailer? Most all have a Sherline scale that they can use with very little effort on their part. If not, maybe you could purchase your own, not only for this fact finding effort but also for future use. When we had our TT, I had one and used it every time we loaded the trailer for camping. That way, I was always absolutely sure of the trailer's tongue weight and I could adjust moving things around to get pretty close to exactly what I wanted to insure my truck wasn't getting overloaded.
However, if indeed the figures come out like you've estimated.....yep, you'll be overloaded very quickly and I forsee a 3500 in your future.
You are one of the very few new post people that have come on here with the knowledge that payload is the main factor when determining what size trailer you can pull. Very good!
Most here, do recommend using 15% of the GVWR of the trailer as the figure to determine the tongue weight (usually called CYA), but I bet the majority of TT's never get to this high of a percentage.
Ron