If I had to guess, I'd say your ground is no good back there, and somehow power is getting to the ground side of the brake light circuit at the taillight harness (perhaps through a bad turn signal switch). When you step on the brake pedal, the power to the ground side is canceling the power from the brake pedal/turn signal switch.

I'd try first disconnecting the connector at the left kick panel and putting a test light on the appropriate terminals (in the dash harness side of connector) as determined from the FSM.

Then also checking the harness from there back with a 12V jumper from the BATT or ACC terminal on the fuse block. See if the lights behave properly...