It is normal to work from the GVWR of the trailer--that would be the ship wt + the CC, which would be around 13700. And it is also normal to assume 20% of the GVWR as the max pin weight of the trailer--2740. So, if the max on the truck is 11300 and 3640 payload, the only things missing are the actual weight of the truck + what you would have in it to travel. You can put 3640 in the truck, but the pin weight might be as high as 2740, which leaves you 900 lbs for people and everything else. You will be right on the edge with the 900lbs you say you will have.
You would also have to be aware of the RAWR and what is already on it when loaded for travel--that, + the pin weight would have to be under the RAWR. And you darn sure don't want to be over the tire capacity.
You need to load up the truck as you would to go camping and go weigh it to get the actual numbers to work with.
And most fivers weigh more than the dry weight when you get it...
Very close, but doable.
Good that you have the 350--lots of folks would have gone with a 250.
2016 Tiffin 40 QBH
2015 38RSSA, traded
2005 TK3 #1869, 10 yrs of memories,
2017 F450 KR--one more Ford is it
2009 F450 4x4-died; 2010 F450-retired