Checking the circuit at the brake switch is where I recommend checking first. Here is the information from the service manual for testing the switch:
DIAGNOSIS AND TESTING - BRAKE LAMP SWITCH
The brake lamp switch can be tested with an ohmmeter. The ohmmeter is used to check continuity between the pin terminals (Brake Lamp Switch Terminal Identification).
SWITCH CIRCUIT IDENTIFICATION
- Terminals 1 and 2: brake lamp circuit
- Terminals 3 and 4: RWAL/ABS module and Powertrain Control Module (PCM) circuit
- Terminals 5 and 6: speed control circuit
SWITCH CONTINUITY TEST
NOTE: Disconnect switch harness before testing switch continuity.
With switch plunger extended, attach test leads to pins 1 and 2. Replace switch if meter indicates no continuity.
With switch plunger retracted, attach test leads to pins 3 and 4. Replace switch if meter indicates no continuity.
With switch plunger retracted, attach test leads to pins 5 and 6. Replace switch if meter indicates no continuity.
As a reference.....terminal #6 on the brake pedal switch is the one next to the plunger.
The brake lamp switch is hard wired to the Center High Mount Stop Lamp (CHMSL) and also monitored by the Instrument Cluster for use by the brake lamp, speed control brake sensor circuits and electronic brake distribution (EBD). The brake lamp circuit is open until the plunger is depressed. The speed control and brake sensor circuits are closed until the plunger is depressed. When the brake lamp switch transitions, the CHMSL transitions and instrument cluster transmits a brake applied/released message on the bus. The Integrated Power Module (IPM) will then transition the brake lamps.
When the brake light switch is activated, the Powertrain Control Module (PCM) receives an input indicating that the brakes are being applied. After receiving this input, the PCM maintains idle speed to a scheduled rpm through control of the Idle Air Control (IAC) motor. The brake switch input is also used to disable vent and vacuum solenoid output signals to the speed control servo.
Vehicles equipped with the speed control option use a dual function brake lamp switch. The PCM monitors the state of the dual function brake lamp switch.
The brake switch is equipped with three sets of contacts, one normally open and the other two normally closed (brakes disengaged). The PCM sends a 12 volt signal to one of the normally closed contacts in the brake switch, which is returned to the PCM as a brake switch state signal. With the contacts closed, the 12 volt signal is pulled to ground causing the signal to go low. The low voltage signal, monitored by the PCM, indicates that the brakes are not applied. When the brakes are applied, the contacts open, causing the PCM's output brake signal to go high, disengaging the speed control, cutting off PCM power to the speed control solenoids.
The second set of normally closed contacts supplies 12 volts from the PCM any time speed control is turned on. Through the brake switch, voltage is routed to the speed control servo solenoids. The speed control solenoids (vacuum, vent and dump) are provided this voltage any time the speed control is ON and the brakes are disengaged.
When the driver applies the brakes, the contacts open and voltage is interrupted to the solenoids. The normally open contacts are fed battery voltage. When the brakes are applied, battery voltage is supplied to the brake lamps.
The brake lamp switch can only be adjusted once. That is at the initial installation of the switch. If the switch is not adjusted properly or has been removed, a new switch must be installed and adjusted.
If the third brake lamp works then the circuit for the brake switch will check okay. As it mentioned above.....the switch also sends the brake pedal switch to the cluster which then sends it to the integrated power module (fuse box under the hood) which then turns the bed mounted brake lamps on. The bed mounted brake lamps are wired directly to the integrated power module.
The most likely cause will be the integrated power module. To be 100% sure....a scanner will be needed that can access the integrated power module to see if any codes and also if the lights can be actuated.