I had to analyze the brake light circuit some time ago. It seems to me that the rear window light was wired along with the left light. I believe the sensor compensates for this "imbalance" with an internal resistor. I may be all wrong but that's what I remember. If correct then the LED's would change the "balancing act" and the warning light would come on.

The whole idea sounds a bit ricey to me.

I check my lights when I am in a strip mall by looking at the reflections in the store window.