As they've said a couple of hundred lbs for the hitch. Add in the DW, rugrats, fido and anything you want to stick in the bed around the 5er hitch.
Looking at a 5th wheel you'll see most of the things your going to haul will be carried in front of the axles. With a lot of this going into the basement just behind the pin. This means that all of that stuff is going to add weight to your pin weight. Not a lb per lb addition but I use 80% as a conservative estimate, it's probably between 65-75%.
If you only haul half of the trailers 3470 capacity you'll add about 1400 lbs of pin weight. Add that to your 2160 and your at 3560 on the pin. Don't even think about 2800 lbs if you max it out.
My trailer has the same gross (13,500 lbs), I wouldn't trust single wheels on the TV for this kind of weight. My old truck had 19.5" tires and would easily carry the weight but the side walls were so stiff any rut would move you around. You'll need 4 tires on the rear axle of the TV for this kind of load. I see a lot of 3/4 tons pulling heavy 5ers but I don't travel close to them. They add helper springs or air bags but most of the time the rear tires are rated about 6200 lbs total.