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I believe the module the dealer is referring to is the box that goes under the rear floor, plugs into the rear wiring harness which I think controls some of the special tow features. I don't think it's required and is different than a brake controller.
That makes sense. That would include the logic which calculates the steering inputs needed, based on the camera input and trailer inputted, and signals to the electric steering system and throtttle how to turn the wheel and accelerate.
I bought the advanced towing package but will likely never use it. But, maybe a future owner of the vehicle might -- or maybe when I become feeble I will?