A lot of people get fooled by the payload rating on the door jamb or in manufacturer's literature. In practice, it's lower and often a lot lower. The best thing you can ever do is take your truck to a scale, weigh it and subtract that from the GVWR. Some will say it's not needed but it's the only way to know for sure. Lot's of things on your truck can take away from the truck's sticker or published payload rating. The actual payload capacity on our 3/4 ton is 1,000 lbs less than what our sticker says and the only thing of any weight we added is a canopy.
Weighing your truck should be your starting point. For the non-believers, go weigh your truck for fun and see what you find.
Warning: don't purchase on the basis of the trailer's factory dry tongue weight. That will go up and sometimes way up. Our factory tongue weight of 518 lbs end up being an actual weight of 960 lbs. That's approaching double the factory listed weight. Depends partly on the various options you add to the trailer since none of them are included in the dry weight. Dealer added items like propane tanks and batteries also drive up the tongue weight. All the stuff you put in the pass-through storage also drives up the tongue weight.
The factory payload capacity for your truck says 1500 lbs. Taking a stab at it, your actual payload capacity might be around 1,000 lbs. And that's before you add DW, the kids, pets, and cargo in the bed. As BarneyS said, the actual tongue weight could be 1200 lbs or more. It looks like you'd be waay over your actual payload capacity TV and TT loaded and ready for camping. Time to rethink your plans. Your horizontal "towing capacity" of 10,200 lbs is plenty though and is not the limiting factor.
You either need to look at a shorter, hence lighter, trailer or a lightweight trailer aka "ultralight", "superlite", etc. A lightweight unit though means lower overall quality, durability and longevity since less substantial materials and construction need to be used to save weight.
Just don't be fooled by published or sticker payload capacities on a TV or dry weights on a trailer. Don't fall for the "1/2 ton towable" label some manufacturers are using. You can get into some major irreversible trouble so easily and quickly. Once the dealer has the RV off their lot, it's all your problem thereafter... No returns for being too heavy.