I assume you have a 4L80E transmission which is actually a great transmission which tows great with very little searching. It will be a little sluggish on really steep grades but just take your time and it will do just fine.
The 4L80 isn't used in a 1500 size truck but was used behind the 6.0 engine in the 2500 trucks for many years.
The wifes 2016 1500 chevy crew cab 4wd 5.3 355 hp/383 torque 6L80E tranny with a 3.42 axle doesn't struggle pulling my 10k car hauler (9460 lbs on the trailers axles) with a blue tractor.
Now if the OP has the tow package (NHT option) it comes with a 3.73 axle 7600 gvwr and 4300 rawr and 8 speed tranny depending on options.
The OP won't have any issues with that smaller trailer in the hills or mountains.
