If the X5 works fine without the trailer connected then it isn't the LCM.

Three thoughts-
1) Have you checked to be sure you've wired the female part of the plug correctly?
2) When you say you have no right side lights is that just on the trailer or on the trailer and the X5? If it's just the trailer, see #1 above. If it's both the trailer and the X5, them maybe you have a faulty trailer light module. The LCM is the box in the right front foot well by the glove box. The Trailer module is back by the spare. You do have one of the factory BMW trailer light modules don't you?
3) How did you wire the electric brake pickup for the P3? Most of us tap off the LCM for the brake input. If that was incorrectly done it may have shorted the right side circuit. Try pulling the plug on the back of the P3 and see what happens. Or you could have a bad P3 thought I've never heard of that happening. Usually someone unfamiliar with the Hall effect wiring on the X5 just taps into one of the brake light wires for the P3 and wipes out the lights. Can't be done the GM way on our vehicles.

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'04 E53, 4.4 Sport
'97 E39 528i
'86 911 Carrera, track car
'96 BMW R1100R