Your 2013 2500 has basically the same 'ratings' as my 2007 3500 SRW
YES it can handle that 13K 5th wheel.
YES it will be loaded but within Axle Ratings, Rear Tire Load Ratings, GCVWR and may be right at or slightly over on trucks GVWR
I tow 14K 5vr (13,873# scaled)........3000# pin wet/just on upper overloads
Never a 'white knuckle ride'
Never a 'OMG' moment----except that time semi truck pulled right into lane and had to slam on brakes.....but truck/trailer tracked straight line
Never a 'Dang sure which I had a Dually' moment
Point is.....
Weight wise truck will be at ratings but will handle it.
Will squat a little bit due to lack of upper overloads so air bag or Timbrens to level up
