A carefully loaded truck (empty), with a carefully setup weight distribution hitch might be able to have a 13K lb trailer attached with 10% tongue weight carefully balanced between the trucks front and rear axles to a stay with in every rating on the truck.
The only way to know is to spend some time at the scale. You must load everything you intend to travel with in the truck and trailer. Unhook the trailer and weight the truck. This reveals how much much weight you have left in the GVWR. That amount cannot be exceeded by the tongue weight of the trailer. You will also see how much weight each axle of the truck is carrying.
Next, attach the trailer with out any weight distribution. Weigh again. You will now know what the GCWR amount is. It cannot exceed the trucks number. You can also see how much tongue weight is added to the rear axle on the truck, and how much it unloads the front axle. This won't really matter because adding the weight distribution system will re-allocate that weight among the axles.
Next, the weight distribution system. Some amount of tongue weight from above will get put on the front axle and some will get put back on the trailer axles, that weight gets lifted off the rear axle.
Lastly balancing the system, you will adjust the weight distribution hitch to maintain about 10% of your tongue weight from step 2 on the truck with out exceeding any maximum weight ratings, FAWR, RAWR, GVWR. You may find that it is not possible to do it. You may find that you're close or...