I know this is kind of an old post, but I was having this exact same problem in my 2008 FX35 Tech Package since I changed from the iPhone 3gs all the way through the iPhone 5, I did not have the problem with my older iPhone 1st gen or the iPhone 3g. Anyway I have finally figured out what was going out and have been able to fix it with a little tweak in the phone.
For this tweak to work the phone has to be jailbroken, I have installed a package through Cydia called "Bluetooth Service Manager", this will install an app in your phone called "BT Services" that will let you enable/disable bluetooth services such as A2DP, handsfree, remote, gaming... on your phone, I thought the problem was that the phone was trying to send its own ringtone through A2DP which the Bluetooth on our cars don't have active, in fact from time to time but only a very few times, the phone ringtone could be heard over the car speakers in a call. Anyway if you enter this app and deactivate the A2DP service, the default car ring will sound with ALL incoming calls, it will still take two or three rings to sound, you will hear it at the same time that the car screen shows PHONE, but as said you'll be able to hear the incoming calls.
Once you deactivate the A2DP service and hit the home button on your phone you will see that the Bluetooth disconnects, you'll have to go to Preferencies>Bluetooth and hit the Bluetooth Handsfree again to make the connection with the car again the first time. After that and if you don't reboot your phone it'll stay with the A2DP service disconnected, if the phone runs out of battery or if you reboot your phone you'll have to run this procedure again.