I had a 2014 Ski Doo Renegade 800 Etec Adrenaline (standard gauge package). I had some issues with mine. Though doesn't seem related to your brake light issue, wanted to throw this weird one out there.
My issue was the sled was running perfectly fine down the trail, and randomly the gauge cluster would go through a shutdown and startup sequence. The needles would fall to zero, and LCD display (fuel, odemeter, etc.) would go blank. Then all lights on the gauges would come on, needles would sweep through entire range, then everything back to normal operation except the handwarmers. If I remember right the handwarmers would not work after this gauge cluster "restart" until I stopped, turned the machine off, and restarted and took off again. The sled mechanically-wise wouldn't miss a beat when the gauges would do this "restart".
The issue was stray electrical current somewhere around the gauge cluster where it snaps into the dash. The factory solution/TSB is a "gauge grounding ring". It was a thin stamped piece of metal (might have been aluminum) that went around the perimeter of the gauge cluster before it snapped into the dash. It was hidden from view when installed, kind of like the shape of a gasket that was behind the outer edge of the gauge cluster bezel. The "ring" of metal had a 1/4" terminal spade on the edge and a female spade connector with a length of wire connected to the ring, and the wire went to battery ground. The ring/wire did not make any electrical connection to the gauge cluster, it was just a "halo" around the gauge body that rounded up stray electricity and sent it to ground. Never had the issue again after that in the few more years I owned the sled.
The Ski-Doo part number is 515177898. Again mine was a 2014 XS chassis, not sure if this fit the 2011 that might have been the XP chassis, not sure if the gauge cluster outer design is different.
And again maybe unrelated to your brake light/ground issue.
