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Sounds like the harness is wired correctly into the correct wires. What type of bulbs (or LEDs) are on the trailer?
When you say you turn on the signals and nothing happens to the trailer lights (no flashing) do the lights on the H3 still flash?
As I stated earlier, the BCM controls the left and right turn signal/brake lights all on one circuit.
I don't know what to tell you without seeing the wiring after the splices such as the U-haul wiring to the connector, and from the connector to the trailer. You could check for voltage at the connector. With park lamps on, power to the green/yellow wires, but when one of the turn signals is activated, the wire to that side of the vehicle should pulse on/off.
Are the LEDs designed for direct replacement for a 12-volt bulb? (These should have enough resistance built in to compensate for the 12-volt bulb that is not being used.) When you say the flashers flash fast, it almost sounds like the BCM senses a dead bulb somewhere in the circuit.
Wish I could be of more help, but long distance diagnosis is pretty difficult.