Reality is, you will probably do fine. My 25 yr old son has a reg cab with a 4.8, 3.42 gears abd the 4l60 trans, he's pulled upwardsof 7K with it, and has no complaints. Does as well or better than my older 6.5td, then again, he has 120 more HP than that motor had, but less torque.
It will pull fine, alibi probably a bit slower and or more revs. I would fine a gear, rpm that the motor is not pushing itself and go up the hills. For my diesels, that has been 2500 rpm or so. For my vortec 350, about 3500-4000. Your 5.3 will probably turn a few more r's than the 350, so 4000-4500, If you need to shift down a gear, but you can go 5500 or so r's, slow to that 4000-4500, and enjoy the ride to the top.
As long as you are not overloaded in axel capacity, have brakes that work on the trailer, enjoy the trip with your kids. When you get over the actual wt capacities of your current truck, look at a new truck. in the mean time, enjoy the time with the kids, they grow up quick!
Another that will allow smaller motors to work better, find a trailer that is more aerodynamic, ie rounded in the front, and has smooth solid walls. The corrogated aluminum trailer do weigh less, but are actually harder to pull, and use more fuel due to worst aerodynamics. SOmething MANY folks do not take into consideration.
Marty
92 Navistar dump truck, 7.3L 7 sp, 4.33 gears with a Detroit no spin
2014 Chevy 1500 Dual cab 4x4
92 Red-e-haul 12K equipment trailer