Bluetooth Issues

I doubt there's a custom solution to your issue, and perhaps you can google similar issues that others might have been able to tackle with jailbreaking and custom codes.

Jailbreak is pretty much your only hope and tweak things from there. So if jailbreak is not something you want to consider, I suggest you to move on and live with it.

 

I would guess that the car kit is wired into the pause area of the radio.

Normally, there is a delay on these to prevent issues such as the ones you are talking about.

What is happening is that whenever the iPhone's bluetooth audio device is engaged, it pauses the stereo.

Basically the options are to alter either the hardware (finding the wire that controls the pause line and fitting a capacitor to it) or alter the software (jailbreak the phone and get it to emulate the actions).