You will be fine. I know this from my real world experience with and older version of your truck. This is not based on guess work and opinions with no relevant experience with the truck and weights you are looking at.
Our truck was a 2012 Silverado crew cab 4x4 5.3 6 spd 3:42 axle.
Trailer is 5400 lb from factory as per sticker and 1660 loaded.
We travel for 5 months every winter. This combination has made round trips from Ontario Canada to the Canadian East coast, to Florida and 3 to the US southwest including 2 over the mountains to San Diego. We average about 10 000 mile towing per season.
The truck has towed that trailer with no problems. Going up 6% slopes we maintain 50 MPH. The engine is revving but that is fine. They are built to do that. 4000 rpm will not hurt anything. My 2012 had 315 HP. The 2016/17s have 355.
You do have to be careful about what you put in the box of the truck. You cannot load it up with firewood but that is irresponsible anyway since that can transmit tree diseases. Most state and provincial parks do not want firewood hauled in.
You will need a good weight distribution hitch. Don't skimp on that. The ones with the separate anti sway bar are not a good choice. We use an equal-i-Zer brand hitch which has worked very well.
Whenever I am towing I use the tow'haul function. It keeps the tranny from hunting and also does provide engine braking on downhill runs. The tranny will actually shift down. I have used manual a few times such as going into and out of a canyon campground we like. The slope is 11%.
The 5.3 with the 6 spd and 3:42 is a better combination than the old 4 spd with 3:73 rear ends. There are charts around that show this.
If you have any direct questions feel free to PM me.
We have sold the 2012 with 100000 miles on it and now have a shiny new 2017 in the driveway with the 8 spd tranny. Can't wait to try it.
Good luck.
