I would do a hard reset. Take the negative off the battery for at least 2 hours. Better overnight if you can. You'll lose your stations etc but worth try.
I would also hard reset the iPhone. Press and hold the power button (the power off slide appears) KEEP holding the power through this until the apple logo is on the screen (takes around 5 -10 seconds after the power slide appears.) This hard resets and clears all cache and often a lot of errors on iPhone. No data or content will be lost so perhaps try this first.
I would also try pairing another phone or someone else's in your family to the car and see if the issue remains with their phone. It is possible there is a fault with the iPhone.