I realize this thread is old, but does anyone actually know what a Diversity Antenna is? The antenna built into the back glass is a radio antenna and a defogger. When you turn on your defogger, The diversity antenna changes the rear glass antenna to a set located in the rear C-Pillars.
So, the radio antenna diversity control module does not have anything to do with your car starting. Now, The reprogramming that was done with replacement of the control module...... That might have helped you on some level.
I just wanted to put this out there because there are going to be people that look this up in the future and should know how silly the noted correction is on this thread.
Please refer to your local dealer with your issues and have a trained BMW technician work on your car.