Make no doubt about it...you are maxing out this truck.
One of my best buds tows nearly the very same package and he has struggled getting every fine detail in order to make it feel solid while driving.
His tongue weight tops out my 1000lb scale, which is exactly 15% of his gross.
The biggest issue has been electronic sway control. Towing at max, has stressed this system apparently and it tries to constantly control the truck making driving miserable.
The solution...turn off electronic sway control. That was the biggest improvement.
Smaller tweaks were changing a few loading techniques for better balence, adding a head tilt washer for a total of 6...which requires a lot rear end lift to pry the bars on the brackets. We also lowered the head a bit more than the book says with good results. These tweaks are very typical, nothing surprising here, but might be helpful for the OP.