My situation is like yours. Battery was dead this morning and I boosted vehicle. I Ran truck for about 10 minutes. Shut it off and it restarted but I could tell it was ragging a little bit. Drove 200 km. Shut truck off for about two hours. Turned key and it barely started. It was then that the "service trailer brake system" warning showed up. Turned truck off to see if it would reset. Turned key on and it barely started. Drove about 20 minutes. Shut truck off and tried to restart about 10 minutes later and battery completely dead. Dismissed the warning.
Drove home 200km and checked voltage for the last two hours to see if there was a drain. No drain evident on battery. Started truck after 2 hours and started no problem what so ever. Also noticed that the "service trailer brake system" warning did not appear when truck started at this point.
I have read the module can be problematic but you are the only one I have seen with battery related issue. I would normally suspect a short in the wiring system somewhere but have to wonder if it is the module by the rear tire. It could be shorting at that location.
Did you solve your problem?