We tow a 337RLS fifth wheel with a 2013 Sierra 2500HD (although with the Duramax diesel and Allison transmission) with a crew cab and 6.5' bed. You don't say what year your Sierra is and towing/payload capacities have increased over the years. The gassers also went from a four-speed to six-speed transmission at some point (I don't know what year). We are slightly under the GVWRR (rear axle rating) with about 3,000# tongue weight. The truck does not squat, the rig sits level, and, with the integrated brake controller adjusted correctly, the rig stopped in much less that 40 feet t 20MPH in a Texas vehicle inspection last week. Your experience in the mountains will be quite different from ours, however, due to the gas engine in your truck. I would suggest getting all the numbers for your truck (should be on the weight and capacities sticker on the door frame), weigh the truck at the nearest CAT scales, and do the math. There are many threads on this and other forums on the topic.
Rob