This is one main criticism I have of GM. They have figured out how to build a reliable simple car. My 2002 Saturn SL has been one of the most reliable cars I have ever had. But in most of their models, they put in too much automatic junk. I guess they think that all their customers are so moronic that they can't even turn on the headlights or the windshield wipers for themselves.
This is where they have almost all their problems. So much for my rant. In the old days, the brake lights were pretty simple. You have some wire that connects the battery to the fuse, the fuse to the brake light switch and the brake light switch to the bulbs. It was either a bulb, the fuse or the switch, and occasionally a broken wire.
Now you have a computer involved. I'm not sure if it uses the body control module to make this a Rube Goldberg affair or it they use a separate DLR module with the automatic light sensor. But in either case, the center brake light was supposed to bypass the computer or anything else so that it at least would light up. Since it doesn't, I suspect that you have a blown fuse that you are not aware of or a broken wire.
Without the wiring diagrams for your specific vehicle, I'm afraid we can't help much. Sometimes you get lucky and someone here owns a similar vehicle and has the factory service manuals, but most of the time it doesn't happen that way.