Hmmm... well, the next thing I'd do is look at the wiring around the
brake light switch to see if something has chafed, you could also check
the bulb failure sensor since many of the lighting circuits go through
it. This is a canister slightly larger than a D battery clipped up under
the edge of the dash on the driver's side.

It would be interesting to note whether the warning lights come on when the
ignition is off and the brakes are pressed. It seems there is a short
between the brake light line (the switched part) and either the accessory or
ignition bus.

Come to think of it, maybe you can get a good clue by pulling fuses to
determine which one will kill the brake lights when the engine is on. Sounds
like the radio fuse is a good place to start.

One other thought, although it seems like a long shot on this one...
whenever I am faced with real electrical weirdness I like to verify the
engine-chassis ground connection. It's notorious for really weird
interactions. My motorhead brother used to work in a shop that specialized
in audio system installations in upscale cars. He completed one, but the
customer brought it back because it didn't work at all. Huh. Works in the
shop, not on the road. Aha! Works when the parking brake is on, but not when
it is released! The engine ground failed and the ground connection that the
sound system relied on was only completed when the parking brake cable was
drawn up tight.

Mike