Hi Rob
When I purchased my RX last year (is a 2013) model I asked the dealer to fit a detachable tow bar, same £1300 cost as yours.
However when I first went to use it I found it had a 7 pin plug on it - called the dealership, who apologised and have since re-fitted the 13 pin at no cost - pretty big job by the sound of it as they booked out 6 hrs labour.
So my guess is the difference between the 7 & 13 pin is the wiring assembly and interfaces (probably not correct terms there!) - surely if they can fit both types to mine they could do the same to yours!!
Sent from my iPhone using Lexus OC