Coming to this party late but with real world experience. My rig is the truck detailed in my sig - a 2011 'Burb 1500 with 3.42 gears, a 2"-ish lift and 32" tires, which are larger than stock and reduce the gear ratio a little further. I'm pulling a 1706FB Winnebago (17' box, 20' total length, dry weight of 2900 lbs, hitch weight specified at 300 lbs). Depending on the trip, our total weight is in the 4500 lb range, with two 5 gallon Jerry cans of gasoline and 30 lb propane tanks added to the trailer hitch. With some stuff in the front storage compartment, a hitch scale is showing 400 lbs at the ball. Whenever my stock trailer battery craps out, I'll be adding more battery capacity, which will mean another 50 - 100 lbs on the ball unless I go lithium.
1) A good weight distribution hitch with integral sway control is essential. Don't cheap out. Even with that, I added Airlift 1000 bags to the Z71 springs I installed after deleting the air ride suspension. I'm able to level the trailer with 40 lbs of pressure in them and the porposing is gone. The Bilstein 5100s do a good job of controlling everything now that I have that issue fixed.
2) The 3.42 does okay but always use the tow mode to keep the tranny out of overdrive - and also improve the shift points for towing. Larger tires will impact low end power and shift points - the rig will work harder because of the reduced gear ratio. I'm thinking of going to 4.10 if I rebuild my diffs even with the small-ish 32" tires. With your four-speed, I'd pull in third no matter what. And add a bigger transmission cooler (see below).
3) Brakes. I've not upgraded mine but am running new pads and rotors. I'm pretty conservative with the bread box in tow but bigger brakes would be nice. They would be a definite addition if I traded for a heavier trailer.
4) I had to address cooling as I bought the truck with a leaking radiator. That said, I went with a 3" dual core radiator which has helped tremendously. I also added a Tru-Cool 40k transmission cooler. Transmission stays nice and cool now, engine stays pegged at 210 no matter what - including steep grades. A Derale might be a better option - they have more mounting options.
Takeaways? Improving the cooling capacities would be my first priority, even pulling a 3000-ish lb trailer. If 6000 lbs, add a brake upgrade. Really good WDH and probably airbags especially if you go with the heavier trailer. Tow in tow mode if you have it, third for sure on your transmission. Slow down and travel two-lane roads and enjoy the scenery on the way...
Happy traveling!