Assuming the 6,500 lb dry weight TT trailer actually weighs around 7,500 lb loaded up, that's the upper end of a properly equipped 1/2 ton truck's capacity, and the sweet spot for a 3/4 ton. The question you have to ask yourself is how far do you plan on towing? If you're going 20 miles from home a few times a year, I think a 1/2 ton will be fine. If you plan on thousands of miles a year (and one cross country trip will do it), I'd go with a 3/4 ton.
Also, forget the 'tow rating' - you need to be looking at it from a payload capacity perspective. That's typically the limiting factor. Look at the sticker on the driver's side door sill. A 7,500 pound TT will have a tongue weight of around 1,000 pounds. Most recent F150s, for example, have a payload capacity of 1,300-1,500 pounds, leaving you with 300-500 pounds of capacity to carry in the truck. That capacity is quickly eaten up by passengers and other gear in the truck.
Yes, there is an HD package for the F150 that can give you up to 2K pounds of payload, but people usually custom order for that. It's rare to find one on a dealer lot.
Again, it all comes down to how many miles you plan on towing. If it's a lot, you want those miles to be comfortable and relaxing.