My camry has this weird problem. The brake lights (both of them) work when I unlock
The car through remote. They work when the key is in ignition and brake paddle is
Pressed. They work then the engine is running in park. As soon as I change gears the
Passenger side goes out and I get the indicator on dashboard. I have checked the
Light actually goes out so its not the sensor for sure.

Initially when the indicator came on I drove around and it actually went away. It
Happened couple of time and once or twice the indicator did not come on as all. But now
Its consistent in behavior.

You have a short in the reverse lights circuit. I commonly find the short in the trunk wiring harness where it bends the most by the left hinge. Good luck finding it. A good electrical guy will pinpoint it under an hour.