The 5.3 is a fine motor. You do need to let it rev to get the power. That is no problem. The 3:42 axle with the 6 spd tranny also works fine. When towing use the tow/haul function and let the truck do its thing. The only time i use manual is on very steep slopes - like 8 to 11 percent.
Your problem is going to be payload. The yellow sticker tells you what that is. My 2012 was about 1600 lb.
A 6200 lb empty trailer will end up at somewhere close to 7400 lb ready to camp. 12 % tongue weight will be over 1000 lb. This will leave you around 600 lb for family and any other cargo.
It will be pretty well impossible to stay under the payload rating when you start at 6200 lb empty.
Try looking for lighter trailers that have bunks. A friend has a Coachmen bunkhouse that is much lighter. He tows it with a Toyota Highlander. I don't know the weights but it is way less than 5000 lb empty.
We have towed the trailer in my siganture coast to coast to coast with a similar truck to yours but with less power from the 2012 5.3
Good luck
