My guess is, that just like the PDIM, any addition to the stock radio, like bluetooth, needs to be "enabled", which only a dealer can do (as far as I know). The GM master website has to be updated to show you have bluetooth enabled, and then that information downloaded(updated) to the radio. The whole thing takes like 5 minutes to do but, unless your radio knows that you now have bluetooth, it just won't work simply by plugging it in.
The only issue I see is that since the stock bluetooth is associated with OnStar and actually physically comes in the OnStar module, there may not be an option to turn it on(if you don't have the OnStar enabled module).
Patrick, I would go to your service techs and have them hook up your car to the tech tool and that to the internet to look at the GM master site and see what options you have available to you. If there is a bluetooth option, select it and save the info. Then update your radio which will tell it to enable the bluetooth. I would take some pictures of this process or some screen shots of it on the pc(cntrl-prtscn) in order to document the procedure for people to take into their dealers. Assuming, of course, that you can get all this to work.
2016 Black 1SS w/ MRC, low-gloss black wheels, auto trans.
1100 - 2/11/2016, 2000 - 3/8/2016, 3000 - 3/9/2016, 3000 w/TPW 4/11 - 3/14/2016, 3100 - 3/28/2016, 3300 - 3/31/2016, 3400 - 4/01/2016, 3800 - 4/09/2016 - VIN: 1G1FF1R74G0169892, 4200 - 4/09/2016, 4B00 - 4/11/2016, 4200 - 4/12/2016, 5000 - 4/13/2016