There are more ratings to consider with RVing than tow ratings and GCWR. Some other ratings is typically exceeded for half tons towing travel trailers well under the tow ratings.
6500 pounds dry will likely be closer to 7500 than the 7000 pounds you estimate. That means 900-1000 pounds of loaded tongue weight. Does your truck have 1800 pounds of payload to handle the TW, people, and cargo in bed? Confirm by locating this sticker on driver's door and note what yours says where mine says 2051:
