[Charli]

Your dealer is probably right.
The GVWR of my 5er is around 13,300 but that's not the weight that will cause the problem. It's the PIN weight. Once I'm fully loaded my pin weight is probably around 3000# and I will probably come in in at 11,800 GVW for the truck.
That said my '11 towed level and always felt in control. When we pack up for our winter trip this year I am going to try to pair back some of the stuff in our basement and with our bicycles on the back I hope my pin weight will come down a bit.
Your GCWR and axle and truck capacities should be fine. And the engine has lots of power. On trailers this big the solution is to travel real light with some stuff packed in the back of the trailer or go dually. That said lots tow heavy units with SRW but that's a hole other debate and there are lots of threads that discuss this issue.