Check stop lamp fault is on. I replaced both lamps - no fix. I purchased a brake relay but don't know which one to switch out

Light bulbs require power and ground in order to work. The ground circuit for the left rear combination lamp is shared between the turn lamp, stop lamp and tail lamp. Because the tail light still works in that specific combination lamp, we know that the shared/common ground for that assembly is OK. The power circuit for both stop lamps (left and right) have the same origin; voltage flows from the STOP LP FUSE (15A) through the DBC RELAY (when the stop lamp switch is pressed) and then to each stop lamp via their respective circuit branches. Because the right stop lamp works, we know that the shared/common fuse, relay and switch are all OK. I'd suspect an open in the left stop lamp power supply branch and would recommending checking for voltage on the green & black wire at the left rear combination lamp assembly to confirm. Regarding the left lamp on the hatch, the wiring diagram indicates these are not stop lamps rather tail lamps only. I'd recommend installing a tail lamp bulb instead of a brake light bulb into that assembly as a first step. You can read more here: https://www.yourmechanic.com/article/how-to-replace-your-car-fuse-box-by-jessica-howe

Is the EBCM energizing the BAS relay to turn on the brake lights ?
The brake pedal position sensor receives both a 5-volt reference voltage and a low reference signal from the electronic brake control module (EBCM). When the brake pedal is pressed, a variable signal is applied from the brake pedal position sensor to the EBCM through the brake sensor signal circuit. The fuse block left rear supplies battery positive voltage to the bass relay switch through the BASS fuse. One side of the bass relay coil is permanently grounded at G402. The EBCM supplies voltage to the bass relay coil upon receiving the signal from the brake pedal position sensor. The switch contacts of the bass relay close and battery positive voltage is applied through the bass relay switch to the center high mounted stop lamp (CHMSL), and to the turn signal/hazard flasher module through the stop lamp switch signal circuit. The ground for the CHMSL is applied from G402. The turn signal/hazard flasher module then supplies voltage to the left and right stop lamps in the tail lamp assemblies that have ground at G401 and G402. For export vehicles the stop lamps receive battery positive voltage directly from the bass relay.

Did you have a qualified repair shop check for DTC'S - diagnostic trouble codes ?
DTC B3903 Stop Lamp Relay Circuit
The electronic brake control module (EBCM) supplies voltage to the BASS relay coil side voltage supply circuit when the stop lamps are active. The coil side control circuit of the BASS relay is permanently grounded at G402.

DTC C0278 Brake Pedal Position Sensor Not Calibrated
The brake pedal position sensor (BPP) is a potentiometer. The electronic brake control module (EBCM) supplies a 5-volt reference circuit and low reference circuit to the BPP sensor. The BPP sensor signal circuit is a direct input to the EBCM. The EBCM determines the BPP based on the voltage from the BPP sensor. The EBCM converts the BPP sensor feed back voltage to a digital signal and puts it on the class 2 communication bus.

DTC C0282 Brake Pedal Position Sensor Circuit Range/Performance
DTC C0283 Brake Pedal Position Sensor Circuit Low
DTC C0284 Brake Pedal Position Sensor Circuit High
DTC C0880 Device Voltage Reference Output 2 Circuit

Without knowing if any of these DTC'S are stored do not know what diagnostic direction to point you in .

Stop Lamps
The brake pedal position sensor is a resistive device used to sense the action of the driver's application of the brake pedal. The brake pedal position sensor provides an analog signal that will increase as the brake pedal is pressed.
The brake pedal position sensor receives both a 5-volt reference voltage and a low reference signal from the electronic brake control module (EBCM). When the brake pedal is pressed, a variable signal is applied from the brake pedal position sensor to the EBCM through the brake sensor signal circuit. The fuse block left rear supplies battery positive voltage to the bass relay switch through the BASS fuse. One side of the bass relay coil is permanently grounded at G402. The EBCM supplies voltage to the bass relay coil upon receiving the signal from the brake pedal position sensor. The switch contacts of the bass relay close and battery positive voltage is applied through the bass relay switch to the center high mounted stop lamp (CHMSL), and to the turn signal/hazard flasher module through the stop lamp switch signal circuit. The ground for the CHMSL is applied from G402. The turn signal/hazard flasher module then supplies voltage to the left and right stop lamps in the tail lamp assemblies that have ground at G401 and G402. For export vehicles the stop lamps receive battery positive voltage directly from the bass relay.

Do you know where the rear fuse box is ? Check BAS fuse 15amp. and the bas relay , should be marked on the inside of the fuse box lid . The rear fuse box is located

Fuse Block - Left Rear
In the passenger compartment, under the left side of the rear seat at carpet
Power and Grounding Component Views
Electrical Center Identification Views
Fuse Block - Right Rear
In the passenger compartment, under the right side of the rear seat at carpet.

If fuse and relay are ok your best would be to take to a cadillac dealer .

Who put the front brakes an wheel hub bearing on the vehicle ? A qualified repair shop ? A friend ? The front brakes an wheel hub would have nothing to do with the brake lights . An the brake light switch is an input to the BCM - body control module .
Stop Lamps
When the driver presses the brake pedal, the switch contacts open and battery positive voltage is interrupted to the BCM. The BCM then supplies voltage to both the turn/stop lamp supply voltage circuit and trailer rear turn/stop lamp supply voltage circuits. The BCM also supplies voltage to the CHMSL though a dedicated circuit.
I suggest you take it back to who did the front brakes , or a quailfied repair shop .
DTC B3445 00: Stop Lamp Circuit - DO you know what a DTC is ? diagnostic trouble code !
Diagnostic Fault Information
Circuit
Short to Ground
Open/High Resistance
Short to Voltage
Signal Performance
Stop Lamp Supply Voltage
B3445 00
--
Stop Lamps Inoperative
Stop Lamps Always On
Circuit/System Testing
Disconnect the stop lamps and test the individual stop lamp assemblies for internal shorts between all terminals.
? If a short between terminals is found, replace the stop lamp socket or circuit board as applicable.
Test the stop lamp supply voltage circuit for a short to ground.
? If all circuits/connections test normal, replace the BCM.

The brake light switch is an input to the BCM - body control module , do you know about BCM'S - electronic module ?
Stop Lamps
When the driver presses the brake pedal, the switch contacts open and battery positive voltage is interrupted to the BCM. The BCM then supplies voltage to both the turn/stop lamp supply voltage circuit and trailer rear turn/stop lamp supply voltage circuits. The BCM also supplies voltage to the CHMSL though a dedicated circuit.

could have DTC'S stored in the BCM
DTC B3445 00: Stop Lamp Circuit
Circuit/System Description
The stop lamp switch is used to sense the action of the driver application of the brake pedal. The body control module (BCM) provides a low reference signal and a B+ reference voltage to the stop lamp switch. When the stop lamp switch contacts close indicating the brakes have been applied, the BCM will apply battery voltage to stop lamp relay control circuit and the center high mounted stop lamp (CHMSL). When the stop lamp relay receives battery voltage from the BCM, the relay coil is energized and the stop lamp relay switch contacts close applying battery voltage to the STOP LAMPS fuse to illuminate the left and right stop lamps.

DTC C0277 00: Brake Pedal Position Sensor Calibration Not Learned
DTC C0297 00: Brake Applied Output Circuit
DTC C0890 00: Device Voltage Reference Output 3 Circuit Voltage Out of Range

In the event that DTC C0277, C0297, or C0890 sets as current in the body control module (BCM) is an indication of a possible internal BCM malfunction or programming error. There are no external circuits available to diagnose on this vehicle with respect to these DTCs.
Your best bet would be to take your vehicle to a qualified repair shop or the Cadillac dealer.

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 where the BCM is located ? If it should come down to it that the BCM needs to be replaced , It has to be programmed an the dealer is your best bet !
Replace the BCM. Refer to Control Module References for replacement, setup, and programming.
There is no relay ! !
Do you know where the BCM is located

Kevin , this is a very complex way of stoplamp control , I will post GM service Description and Operation of how they work an let you deside if you want to tackle this ! Do the center high mount brake lights work ?
The brake pedal position sensor is a resistive device used to sense the action of the driver's application of the brake pedal. The brake pedal position sensor provides an analog signal that will increase as the brake pedal is pressed.
The brake pedal position sensor receives both a 5-volt reference voltage and a low reference signal from the electronic brake control module (EBCM). When the brake pedal is pressed, a variable signal is applied from the brake pedal position sensor to the EBCM through the brake sensor signal circuit. The fuse block left rear supplies battery positive voltage to the bass relay switch through the BASS fuse. One side of the bass relay coil is permanently grounded at G402. The EBCM supplies voltage to the bass relay coil upon receiving the signal from the brake pedal position sensor. The switch contacts of the bas relay close and battery positive voltage is applied through the bas relay switch to the center high mounted stop lamp (CHMSL), and to the turn signal/hazard flasher module through the stop lamp switch signal circuit. The ground for the CHMSL is applied from G402. The turn signal/hazard flasher module then supplies voltage to the left and right stop lamps in the tail lamp assemblies that have ground at G401 and G402. For export vehicles the stop lamps receive battery positive voltage directly from the bass relay.

There could be a code stored in the EBCM - electronic brake control module - DTC B3903 Stop Lamp Relay Circuit
DTC C0278 Brake Pedal Position Sensor Not Calibrated
there are several other codes that pertane to the brake lights . you could try Replaceing the bas relay !
Fuse Block - Left Rear
In the passenger compartment, under the left side of the rear seat at carpet.

check the lt blue wire at upper lamp,(under trunk lid trim panel) for 12 volts when brake is applied. if no voltage,check same color wire at brake lamp switch for voltage.(by brake pedal).if no voltage replace stop lamp switch asm. if voltage is detected at rear by lamp then replace CHMSL asm.(center high mount stop lamp). this lamp is made of many LED's,if some work and some don't you will need to replace lamp asm. can be repaired by a trained GM tech, possibly. there is no seperate fuse or relay for this lamp,only a stop lamp fuse.

https://s8.postimg.org/5js3uvryd/park_lamps.jpg
Click on the link.
Did you check fuse circuit for voltage? Use a test light with headlamp switch on to check for voltage. See the park lamp relay, I'm not saying the relay is faulty, but, you might be able to use jumper wire on load side of relay, if lamps work, the problem is on control side of relay, for some reason.
The load side of relay is that line with arrowhead. There could be other issues.