Your truck has a GCVWR of 13,500. If your TT has a dry weight of 6500lbs it probably weighs around 7500lbs loaded. With a TW of 900lbs. Your truck is maxed out on GVW.
Assuming your truck has a GVW of 7200-7350lbs and the TT weighs 7500lbs, then you're over GCVWR by 1200lbs or so. The Tow package includes, receiver, extra tranny cooler, upgraded radiator and wiring harness.
You must've sprung a leak somewhere.
Are you 100% sure you have the tow package? Why would the dealer tell you to install another tranny cooler when the tow package has what's needed?