I suspect there's some resistance upstream of the brake switch and it's creating a voltage divider when you operate the brakes.
Looking at a wiring diagram there's nothing on the same feed, via the top fuse in the fuse box (white to green). Tracing it back through the ignition switch (first suspect) there's a circuit via a line fuse that feeds the instruments: Ignition & oil lights plus voltage regulator. There might be an earth short, but you'd expect the lights & gauges to show something too. Another branch off that line fuse operates the fuel pump relay, the emissions purge valve (switched to ground at ECU) and connects to pin 11 on the ECU, presumably just as an ignition on signal - that would be my 2nd suspect, including everything back to the line fuse. You could bridge the fuel pump relay terminals to eliminate that - then swap in another relay to test for relay or wiring if required.
...and,
The reverse lights do share the brake light supply & have the same load, so they ought to cause the same fault.
Ethel, thank you for your reply. Are you referring to the attached schematic diagram? This is the only diagram I have that shows brake lights, so I hope it is the correct one. I did notice the connection from the white wire to the green wire via fuse C6 (this fuse is okay, I have inspected it). There is however a notation in the schematic that I don't fully understand. The green wire has a connection point with a black arrow where it says "see power distribution (A)". The white wire has the same. Does this mean that power is coming into joints SJ5 and SJ7 from the power distribution schematic? Or does the direction of the arrow indicate the direction of the current? I'm not sure how to read this.
You might be correct that the reverse lamps would cause the same issue (engine stalling) as the brake lights if it is a bad connection upstream. Unless the green wire has a bad connection at SJ7 going into the switch, then the problem is limited to the brake lights only. To be honest: I have not tested what happens if I put the car in reverse, thus far I have solely focused on the brake lights. Now that that has a workaround with the LEDs I might be able to put the car in reverse. Side question: where would SJ7 be located in the car? I don't have drawings that show where junctions and connectors are located.
If the current is flowing from the brown wire, via the white wire, to the green wire that supplies the brake pedal switch I would need to search for those wires. And probably check the ignition key in the steering column?