I like your way of thinking, but I'm not sure the turn signals are managed by the BCM. Now granted I am looking at the service manual for my 2005 Tahoe, but a lot of the systems were pretty similar for the following generation. I don't see any turn signal wires going in or out of the BCM. So check the wiring description for the BCM harness and make sure the turn signals are being controlled there. In the previous generation the signals and emergency flashers have their own module.
I know I am late here but no, you are very incorrect. Starting 2007-2014 the BCMs function very differently. And CANBUS was in its second iteration then if I'm not mistaken. I probably am with the CANBUS thing. But not the BCM. Older models had a turn signal switch and a relay. Really simple stuff. Then, for some reason, prob because they wanted the blinker sound to sound
less mechanical, less like a relay jumping on-off-on-off. Now it's tick-tock-tick-tock through the speakers of your car. I had a Cavalier one time when I began 12v installing professionally, and I despised GM so much for adding their factory CANBUS interface that kept all factory communication with the BCM, but at the point of hardwiring it and bypassing that CANBUS interface that made that install harness almost as expensive as some of our entry level single din radios but themselves, we were talking about maybe the blinkers not making any noise but operating fine, and
maybe your check engine light and/or airbags
might not work but on my own Cavalier, I
refused to pay that $90 for that harness back then because that's how much it was and most shops weren't hardwiring any longer. I sure as hell was gonna do just that. I got all these cool new logins from my new job at the shop that gives me entire wiring diags
plus their locations and later pics, and how to do stuff. So I decided I'm just gonna grab a fluke and a 9v battery to pop the speaker wires against what these diags told me the were and also verified in other ways that I wasn't blindly doing too much. But yeah. way different tech. It seems as though that number 2 in front of what year we're in made it so that nothing in vehicles was ever allowed to remain mechanical, or at the very most, minorly technical with much smaller micro relays and such inside and some caps and resistors but not much more. Now each system is its own freaking computer. Man i sound like an old man right now. Some would say I am at 40. I feel old but same. If anyone was wondering.