Towing 13,000 pounds you will be much happier with a diesel 3500 SRW. 2500's just do not have the CC for trailers that size. Later today I am going to weigh my 2015 3500 SRW ready to tow. Given that it weighs right at 8K without me, the boxes, and hitch, I am guessing I will be at 8600-8700 with two of us and set up for the snowbird flight. That is going to leave 3K for pin weight, and if I want a larger trailer I should have gotten a dually! Lightly load with the trailer the rear axle was at 5920 with a RGAWR of 7K. So I have more over head on the rear axle than the GVWR of 11,700.
With a 2500's 10K GVWR pin weight gets you over very quickly.
Chris