I've been through this process. I bought a brake light that fits on my Shad luggage rack under the impression that it was LED based. It wasn't and when I connected it across the existing brake circuit it triggered an error and shut down the circuit. I then removed the bulbs and their holders from inside the light and fitted an LED strip in their place. This worked OK, presumably because the small extra current was within the tolerance of the Canbus settings for that circuit. I later fitted a HexezCan for other reasons but decided to connect the additional brake light to that as I also obtained some brighter LEDs and thought they might trigger the Canbus.
I think using a relay with the brake circuit will not be practical as the same circuit also drives the rear light, but provides reduced power for this and just increases the power to maximum whenever the brakes are activated. I believe it does this not by supplying a reduced voltage, but by a method called pulse width modulation (PWM) where the circuit is rapidly turned on and off, with the off interval set to give an overall average level which gives the correct brightness. This could cause the relay to also try to switch on and off very rapidly to the point where it would probably rapidly fail.
The reason that you don't see the PWM as a flashing rear light is that the on and off flashing is too quick for the human eye to register, so you just see an average brightness. A flashing effect is sometimes visible on video footage because video essentially consists of a series of still pictures or frames, and depending on the relative frequencies of the the video frame rate compared to the PWM frequency some frames may catch the LEDs turned on and others might catch them turned off.