My new daily driver is a Silverado 1500 with the 5.3L, 6-speed, and 3.08 gears. It's a whole different animal from a 3.08-geared truck of 10 years ago.
I occasionally tow a 4000lb+ tandem-axle enclosed utility trailer about the same size as your new travel trailer, and this truck tows that trailer down the highway in 5th gear with the tow-haul mode activated, and rarely downshifts.
Tow-haul on the 6-speed appears to lock out 6th gear.
You are not going to have any trouble at all, but be aware that if you decide you like this camping thing, you will get the itch to "upgrade." You do not have any wiggle room to upgrade with your current tow vehicle. It is capable of holding the tongue of a bigger heavier trailer up off the ground, but you will not be happy with the performance, and your truck's transmission will not be happy pulling it.
