The brown wire on wiring harness # 119178KIT or # C56175 for your 2007 Suzuki Forenza is for the running lights so if it is not connected then that would be why the running lights are not working on the trailer.
So the question now is why does connecting the brown wire cause a short? There could be more than one wire that carries a running light circuit. Make sure to test the wires with the included circuit tester to figure out which wire(s) carry the running light circuit on your vehicle.
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