My car doesn’t do Bluetooth so I’m not familiar, but can you not just not connect it to Bluetooth if you’re not using it?
There are a bunch of settings in the Android connected devices settings, maybe changing the device type would help.
If it’s Android Auto, I can see that there’s a toggle for notifications, but that probably just turns them off.
I’d also see if there’s a way to update the software in the car. Maybe it’ll improve the behavior. Or even just give up on Bluetooth music and use a separate adapter that behaves better.