"Factory fit" towbars aren't always fitted in the factory. That's definitely the case with Lexus! They're just fitted by the manufacturer, or even dealer, before you get the car.
What you have is a towbar module\ecu that connects to the car's CAN bus and then has the wiring to the socket. The factory fit one will be an MB module and one that can deploy, retract and respond to the status of the deployable towbar. If the car isn't commanding the deployment, or querying the status of, the towbar then the module may still do exactly what you need. It all depends on the firmware inside the module, if for example it needs the "activate" command to turn on the 13 pin socket then it won't work. If the 13 pin socket is live all the time anyway then you could get away with the same module. If the same module is used for deployable and non-deployable towbars then you should be fine.
An aftermarket module (one that doesn't need coding in) is passive. It just listens to the CAN bus for commands to operate the indicators etc. It may inject a "bulb blown" message but it's not talking with the car with regard to the towbar being deployed or not deployed.
The people who fitted your towbar will have obviously disconnected the wires to the towbar (rather than the 13 pin socket). There probably will only be 2 or 3 wires, one may be GND, one to power the motor and one for feedback. It's the feedback one that needs the fudge. Although if the module is really clever it may also be monitoring the current to the motor.
Unfortunately none of the above is definitive, it needs further investigation.