I attached the drawing for the BSM/Lane Change Assist system. You are correct, the same sensors (J1086 and J1087) are responsible for Rear Cross Traffic alert. Per the drawing, this system is not connected to the BCM. I retrofitted BSM and it's a relatively isolated system, as compared to others on the car. Power and ground of course, and connection the gateway (extended CAN, J533).
As I mentioned, power to both indicators (VX66 and VX67), is provided by (one of) the monitors/modules itself, J1087. You have proven that it monitors the indicators, probably looking for the specific resistance provided by the factory LED indicators. You could (maybe) tap into the voltage line going to each indicator and use it to switch a relay that separately powers your housing mounted indicators. You would need to keep the factory ones connected though, and probably protect the existing system with a cheap diode. That way the modules see what they want so the system can be activated. And you get what you want (no fading of the housing indicators though).