I don't have a solution for you, but I happened to be reading an auto electric book that had something similar. Depressing the brake pedal caused several other electrical glitches to occur.
The Short To Voltage Story.
From the text: "Because most of the trouble occurred whenever the brake pedal was depressed, the technician decided to trace all the wires in the brake light circuit. The problem was found near the exhaust system. A small hole in the tail pipe (after the muffler) directed hot exhaust gases to the wiring harness containing all of the wires for circuits at the rear of the truck. The heat had melted the insulation and caused most of the wires to touch. Whenever one circuit was activated (such as when the brake pedal was applied), the current had a complete path to several other circuits. A fuse did not blow because there was enough resistance in the circuits being energized so that the current was too low to blow any fuses."
It doesn't concern a Lexus, obviously, but maybe something similar is going on.