2013 Chevy 1500 5.3 Anyone Tow with this?

IIRC the RWD models have a 3.23 std, or a 3.08. The 3.23 is actually similar final drive ratios to a 3.42 geared 4wd. Reason being the tire diameter is about 1" taller for the 4X. Which equates to about half a gear. 
Also from 2013 to 2014, a majority if not ALL of the manufactures gave their gas rigs a BIG HP gain, along with more gears generally speaking in the transmissions, and better MPG! Which equated to better towing ability performance wise too! So a 3-4000 lb increase for that motor is about right. ALL in reality this means, one of the standards to make a given GCW is for a min speed of 40 mph on a 5% grade. your truck might be going 45 with said trailer, meanwhile the 2014+ truck will be going 50-55mph due to the higher HP, closer spec gears in trans, ie a 6 or 10 sp instead of a 4, 5 or 6 sp trans. 

That is not to say you can't run down the road at 15K gcw with your current rig. Assuming you have the correct paid for tags, you would be legal at 15K lbs too! At least here in Wa st, the State Patrol LEO and CVEO doesn't care what your door sticker, manufacture warranty, performance weight rating is! They only care that you meet the "Federal Bridge Law" specs, along with having enough paid for tonnage, ie tax paid! for running down the road at XX lbs! I've been pulled over upwards of 150% of my manufactures GVW in my dumptruck, all but one time I was under the paid for tonnage. I was 1200 over, they gave me a 10 day up the gvw by 2000 lbs. Weight tickets are in reality like parking tickets, they generally speaking do not go against a driving record. 
BUT, if you were pulling without brakes on your trailer....that will hurt! As you will not stop in the field test portion of brakes working. SO the rig will be red tagged, ticketed for poor performance, ie not meeting minimum spec. Only way off the road to "FIX" the brakes is a tow via flat bed, or worst case, low boy trailer or equal mega expensive tow! This ticket will go against your driving record! The overweight ticket will generally speaking not go against your driving record. 
After ALL the stuff I just typed. You can figure out the ratio you have by looking in the glove box for the RPO codes. IIRC a GT4 is a 3.42 ratio, GT3 a 3.08, not recalling the 3.23. BUT, if you look up GM RPO codes via google or equal. You will find some sites with the codes. Many break out via what the code is for, ie axels, or engines, or transmissions, interior options....... If you don't have the original paperwork with options, its not too hard to figure out the options you had on that truck new. Some of the options may have changed.....like someone with a 2014 to be left anonymous, put 285-65-18 tires instead of the stock 265-65-18 tires in 10 ply version vs SL. A 2" lift kit on par with the Elevation pkg. running boards and bed cover...... Nothing wrong with any of this. Just saying the codes may or may not be correct. We will not discuss Anonymous's sons 05 reg cab 1500.....motor mods, 4' lift, swap out an 8.5" 3.42 geared RA for a 9.5" 4.10, front axel to 4.10, 33.5' tires instead of stock 31.5........

Any way, quit worrying about the 1500 you have in your hands, it should do quite well pulling the trailer you are going to pull. NO, it will not be as fast as the deezal Dodge/Ram 1 ton you have or had. Nor as fast as my now sold and gone DA CC DW rig I had. It will get their safe and sane, as long as the driver has done due diligence to make sure lights, brakes, trailer is in balance for and aft for best towing, side to side is equal in weight. My max for my truck is 2x to pushing 2.5x the RA amount, so 2 x 4000 = 8000 lbs of trailer max. SHort distances, ie local, I'll push 2.5. Long hauls the 2x ra is more comfortable, as long as I have the power, trans gearing to keep myself above min interstate MPH which is generally speaking 40 mph! THat is on a 4-6% steepest freeway grades. Local roads where there is no max grad %, ALL bets are off. I've been on some in the 25-30+% relm, one can get stalled out with out the proper drivetrain. Max towing spec used is 12% currently!

Marty