Hi Travis , lets see if we can help you ! First off here is a description of how the brake light circuit works , you need a good understanding of how it works before you can fix it.
The 15-amp STOP fuse located in the body control module (BCM) supplies battery positive voltage to the normally open stop lamp switch. When the driver presses the brake pedal, the switch contacts close and battery positive voltage is supplied to both left and right stop lamp assemblies and the center high mounted stop lamp (CHMSL), through the BCM. The stop lamps are grounded at G401 and G403.
Do you know what a BCM is ? The body control module ! An electronic module .Just like the PCM/ECM - engine computer but for the body thing's ,light's , anti-theft system ,power door lock etc..
The body control module (BCM) has a bussed electrical center integrated into the housing. Body control relays and fuses are installed directly on the BCM which simplifies wiring and junction blocks on the vehicle. The body control system consists of the BCM and its associated controls. Battery positive voltage is provided to the BCM from the instrument panel (I/P) BATT 2 fuse in the Underhood fuse block. The module grounds are wired to ground G201 and G301. The BCM is wired to the Class 2 serial data communication bus as well as discrete input and output terminals to control the functions of the vehicles body .
With starting problems an the brake lights problem you may have a BCM problem , could have DTC's - diagnostic trouble codes stored in the BCM . This would have to be checked at a GM dealer or some other independent repair shop with a professional grade scan tool .
The ignition switch is an input to the BCM , when it receives a start input the BCM energizes the run / crank relay an other things !
Relay Controlled Power Mode
The BCM uses the discrete ignition switch inputs Off/Run/Crank, Accessory and Run/Crank to distinguish the correct power mode. These circuits are also routed to the Run/Crank and Accessory relays for relay controlled power feeds during the appropriate power mode.
The relay uses a Hot At All Times B+ power source derived from the underhood electrical center. The Run/Crank relay supplies a power signal to the following circuits when the a Run or Crank power mode is selected:
Backup lamp circuit
Electronic brake control module (EBCM)
Engine control module (ECM)
Ignition control module (ICM)
Park neutral position switch
Transmission circuit 1 and 2
This is not something for a do it yourselfer , this is dealer or like I said a independent ASE certified repair shop .