Same problem here. iPhone 13 Pro and 2012 Citroen C4. I turn the car ignition on and the iPhone connects (both media and hands free) but after 10 seconds or less the hands free disconnects. I have tried everything but nothing works and after all i'm assuming that the problem is from the iPhone bluethooth because i've tried with my friends android and works fine.
I have checked with Citroen and theres no updates left to do in my car and my iPhone is updated with the last version (15.4).
So, only Apple can help with this and it's a major fix needed because it's a really important topic since people can have car accidents using phone while driving...