I'll preface this by saying that I have not seen any wiring diagrams or technical papers/flow charts on this
But from what I can tell the towing control module (I think the factory term for it is the TRM) sits on the right rear fender and subsequently handles the output for the 7 pin connector. When it detects that a trailer is connected (I'm not positive of the logic sequence used by Ineos, but other brands use a load detection across any of the light circuits) it should engage several different logic changes in the vehicle. Again, I haven't seen any specific technical charts on this to confirm, but based on other vehicles these changes could include adjustments to the ESP/ABS algorithms (for sway control), turning off the rear parking sensors, etc. The programming in this module dictates whether the vehicle has the 7 pin (US) or 13 pin (I think this is right?) Euro connector and of course what/how the outputs are triggered.
My assumption on which module it is exists primarily because it has a wiring harness that runs out of the vehicle through the adjacent grommet, and not every photo I've seen has had this module (it is unrelated to having the relay for the NATO plug). Regardless of where it is, it most certainly exists, as it is the primary module that was received new software with the US update to allow the hitch installs.
In the case of retrofitting a towing hitch after production, the most likely way would be to install the physical receiver, and then grab triggers for the lighting functions from the taillights (I say triggers because you would want to relay all of these as to not place additional load on those circuits); I believe there are kits that are sold to do this to vehicles that don't have factory trailer connector wiring, of course the vehicle would never "know" there was a trailer. I'm not sure what the possibility is of retrofitting the module, at the very least I imagine it would require dealer programming to code it to the vehicle/region, and that is if the harnesses are pre-run to the module. Then of course you would require the rest of the wiring from the module to the trailer plug.
From what I have been able to deduct after way too much internet searching, while the "towing" module may (and speaking from experience does) change some of the driving performance and operations of the Grenadier itself, it does not allow you to adjust the level of electric braking going to the trailer.
In other words, if you pull a trailer without brakes, connect it electrically through the factory 7 pin connection, the module that is in the Grenadier that came from the factory with the tow package, will sense that you are towing, and make adjustments to the vehicle's operation. It will power the trailer lights and signals. If you pull a trailer with electric trailer brakes, the Grenadier will make the same adjustments to the vehicle's operation, also power the trailer lighting, but it will not engage the electric trailer brakes.
To engage the electric trailer brakes, you need a third party brake controller.
That seems to be the consensus and seems to be supported by the fact that Ineos provided that wiring even for people (like me) who ordered those with the factory tow option.