If the CTS BlueTooth is standard, it should 'pair' up without any difficulty.
What results did you get when you tried to 'pair' the car device to the iPhone, i.e., what message did you get on the iPhone regarding why it would not pair?
There is no Bluetooth in the CTS. OnStar and Cadillac have an agreement that does not allow Bluetooth to be factory installed. You must use an after market Bluetooth device.