i just popped the back side and put a multimeter there...

with AHM out, the pins 10 and 12 show just the floating voltage (without an O-scope hard to tell if there is a pattern in the voltage) about 0.5V regardless of the front switch - it can be on or off...

with the AHM back in, the pins 10 and 12 are the SAME - voltage is the same, about 0.5V, again regardless, if the switch is on or off...

however, when the AHM module is back in, the pin 3 (which is blank on the plug but the module has the pin) has voltage of about 14.25V (battery voltage) with switch on and about 0.5V with switch off...

this tells me that the LCM is not sending the rear fog voltage out to the pins 10 and 12, but sending an iBus telegram to AHM telling it that the rear fogs are switched on at the front switch, and the AHM generates its own power based on that telegram - in the factory setup, the rear fogs and the reverse lights are turned off on the car when trailer is connected and the rear fogs and reverse lights are activated on the trailer itself..

so...

the actual power to the rear fogs is not coming from the LCM

without the AHM, there is no rear fog lights

with the AHM, the power still is not coming from the LCM

with the AHM, the iBus telegram from the LCM tells the AHM to activate pin 3 (as in the original write-up) which activates the rear fogs on the trailer...

so, now is a good time to venture into the NCS Expert to find the rear fog lines and find out what values should be...
from what i had read about the AHM module, it is an extension of the LCM, and performs its own monitoring and powering the lights...

I am sure that the LCM has the fogs turns off in the programming...

the day is still young - there is time to keep digging...