The stickered weight of 1070lbs is your trucks actual payload rating. That is the amount of weight you can add to the truck before exceeding your GVWR. The truck doesn't care where the weight is added, in the cab, in the bed, or hanging off the back as tongue weight. It all counts against payload. Mind you, if you did add it all in one place, say all 1070lbs in the bed, you would likely exceed other ratings like rear axle and tires too.
The Jay Flight will put you over some ratings. Up to you to decide what you're comfortable with. At 6100lbs dry it will probably be closer to 6400lbs with options added (batteries, propane etc). You'll probably be closer to 7200lbs loaded up which would put you in the 900lbs tongue and hitch weight range. Add a couple of passengers and you will be over your payload rating. Will it kill the truck and be dangerous? Not necessarily. If you stay under the axle and tire ratings it'll probably be fine but you'll always be pushing the limits and having to watch what you pack. Not really a comfortable feeling when you're trying to get away to relax.