Somewhere on this site there is a PDF of the trailer hitch installation for the E53 (I downloaded it some time ago).
On the last page of this document it is has a wiring diagram.
Although this document is written in English for the most part, the wiring diagram itself is in German which isn’t a big problem as you can translate certain words but it contains what appear to be abbreviations that I have not been able to translate.
What I have determined is that pins 9, 10, 11, 12 & 14 on the X609 connector appear to be the outputs for the trailer. I believe pin 9 is for the tail lights by process of elimination as it is the only one that doesn’t have a left and right.
You will need to experiment to find the others from the remaining pins mentioned above.
One issue you may come across is that the stop/brake light will probably have two outputs as the US and Euro trailer connectors have two pins for this (left/right).
You may find using one of these will be sufficient to drive the trailer brake lights although I don’t know about the current capabilities of one output from the module supplying all the power for the brake lights.
This method may also flag a bulb failure in which case you will need to connect the two brake light outputs from the X609 together via diodes. This would mean connecting the anode of 2 suitable diodes to each of the two brake light outputs, then connecting the two cathodes together and using this to connect to the brake light circuit on the trailer.

I’m rather keen to know how this goes as I’m about to undertake fitting a towbar to my own E53. Now… I wonder if ECS have any of those kits left….