They make it sound so simple
I have been trying to solve the same problem on my 2000 American Eagle, I figured should be easy as pie!
There already is a 7 pin round plug on the rear so I just need to install the brake controller.
I have one that I installed on our Toyota Venza (2011) and it removes really easy so I figured I'd use the same unit after ordering an extra mounting kit, but I ran into the same problem so I called on the guys where I bought my last trailer hitch and controller, a Tekonsha P3.
They said you can but seriously recommended against controlling an Electric system from an Airbrake system vehicle with just a standard controller, there are some other options but I decided to go with a wireless remote handheld system they recomended, also by Tekonsha the Prodigy RF list $449.95 best price I've seen in US was on E trailer.com for $308.- I had to buy in Canada where it was considerably more.
The big advantage with this system is that you can move it easily from vehicle to vehicle. Even the trailer unit is simple to transfer to another trailer if needed, only 4 self tapping screws and you're done.
I know this does not exactly answer your question but this may be a better solution as it was for me.