Sorry if that wasn't helpful. I admit that am unfamiliar with third brake light units that also include (somehow) running lights and TURN function. That 'signal' out of the brake switch should absolutely produce 12V whenever the brake pedal is depressed though. It's just a switch, with battery voltage on the other side, after all. What you have should have some instructions. You know... if all else fails... Or consider getting a simple strip light, without the additional functions. That should work.
As an electrical engineer, this electrical stuff always seems easy,... until it's not, that is!
-V