Welcome to JOF
Since we all have different TV/TT loading habits and towing expectations, IMO these factors should be taken into consideration in respect to your question.
Do to the limited Payload Capacity of most 1/2 ton TV's, the probability is very real that your TV may be max'd out (at GVWR) even under conservative 27RLS loaded conditions. Please note, the TT's "loaded" tongue weight is supported by your TV, thus is subtracted from your TV's "available
" payload capacity.
The 27RLS has a published UVW of 6,030lbs, but the actual "shipped" from the factory UVW will be closer to 6,500lbs (yellow sticker on TT identifies actual UVW). So (6,500lbs) + (100lbs dealer battery, etc.) + (800lbs your TT cargo, no fluids) = 7,400lbs
under moderately loaded conditions (27RLS GVWR is 8,600lb).
The recommended loaded tongue weight range for a 30ft TT would be 13% to 15% of the loaded TT weight...., in this case @ 7,400lbs = 962lbs to 1,110lbs
To have a better understanding if your Silverado would be towing under, at, or above it's specified GVWR, take your TV to a CAT scale under "loaded" conditions
(full fuel, passengers, cargo) and weigh it...., subtract the CAT scale weight from your TV's specified GVWR noted on the driver's door.., the weight remaining is your "available
" payload capacity. The CAT scale weigh-in will take only a couple minutes and about $9.
Knowing the available payload capacity based on your configured/loaded Silverado, and factoring in your TT loading habits, IMO will eliminate a lot of the guess work in answering your original question, "Will my 2013 silverado 1500 Z71 5.3 with 3.42 gears pull this trailer ok". Nothing wrong with towing at a TV's GVWR, but depending on one's personal towing expectations this may/or may not be a factor if one tows frequently in the mountain.
Hope the helps