It's easier than what you propose. Varied camera and GPS applications have suction cup arms that can be used to mount a camera in the rear window. Go to the GoPro section at Best. You will find a solution if there is not one in your garage parts box.
We are using that method. It makes for an easy install. In our application, we do not have the 12 volt
source and use a jump start battery that has the cigarette lighter style socket. Lasts several days before needing a recharge.
The problems are -
The rear window collects moisture when it rains. The air flow at the back of the trailer does not pass over the rear window. The higher mount near the clearance lights is in that air flow.
The suction cup will release from time to time. Not really an issue as it works quite well and is easy to correct.
The window is a source of some distortion. There also can be condensation on the inside that distorts the image.
The view down is blocked somewhat by the internal position. The view behind is what we find to be of value.
Certainly, is an option to get a camera in place quickly while you review your options.
Travel safe. Pat