The wiring diagrams (particularly for North American sold cars) are the the rather helpful, thick Bentley manual (it's helped me for my Euro E53 as well).

For the tow bar / trailer hitch, what would be helpful to assist you - what wiring harness and system do you have? Westfalia (which is the OEM mfr for the E53)? You DO have the specific module (its in the largest compartment where your battery is located)?

Can we assume that your car's brake lights operate correctly (off and on) and that only the trailer is the problem? What results do you have when you use your running lights (1st position) or headlights (2nd position) or high beams?

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2002 Euro X5 (E53) 3.0 diesel (M57)
5 speed manual with UUC short shifter.
Pre-xDrive NV125 transfer case (chain & gears).
Factory Xenon headlights, Spyder LED tail lights
Updated vortex crankcase breather.
Motorsport dead pedal, factory fire extinguisher, factory first aid kit, factory F&R mud flaps, factory PDC

Bilstein B6 shocks F&R
Redline oil has replaced "lifetime" fluids (F&R diff, manual transmission, transfer case and P/S).
Dimple magnetic drain plugs in all compartments.