Not having the actual numbers in front of me I would say your take is correct. When planning for hitches and tires and TV capacities it is best to always figure the GVWR of the trailer, not the unloaded or partially loaded numbers. As I recall, this is what happened to me. The weight bearing tires were over when compared to the dry weight but not enough for the gross weight.
All that said, some of the weight is placed on the ball of the TV so it isn't as clear as you might think. If you have the trailer, take it in and have it weighed. You will then know how much weight is sitting on the axles. When you get new tires make sure you have a buffer above the actual weight of your loaded trailer.