Your concerns are valid. If your calculations on the truck are correct, you have 1100 lbs payload to work with.
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If you subtract the actual truck weight from it's GVWR, the remainder is your payload.
From your payload you need to subtract loaded tongue weight, 100 lbs for the weight distributing hitch, the weight of your occupants, and the weight of any cargo in the truck.
With 6246 dry weight, the trailer will weigh (average load of dishes, pots and pans, camping gear, BBQ, tools, groceries, water, and bedding, etc, is 800 - 1000 lbs) 7000 - 7300 lbs loaded. Average tongue (hitch) weight is about 12.5 percent of loaded trailer weight. That would give you tongue weight between 875 and 912 lbs. The 340 lbs, you are concerned with, will more likely be under 100.
