Notification Sounds and Bluetooth

I have recently moved to an iPhone after years and years of using an Android phone. It's a 12 Mini and it's running iOS 15.5. I love this phone and it works beautifully, but there is one thing that has been bothering me.

My phone automatically connects to the Bluetooth on my car sound system, as it should. However, when it is connected, all sounds are routed through the sound system. As a result, if I am streaming, I get notifications played over the sound system. If I am not streaming from my phone, however, but rather listening to the radio, I do not hear notifications from my phone, with the exception of emergency alerts, because all sound is being routed through Bluetooth.

On my Android phone, notifications would always play through my phone. On this iPhone, I can configure Waze to play the spoken directions through the phone while I am streaming music from the phone through the sound system, so I know it is possible for the iPhone to have two different audio outputs through two different devices (Bluetooth and speaker) at the same time. However, I can find no documentation anywhere that tells how to make my notification sounds always play through the phone.

I don't want to turn off Bluetooth, because I need it up and running in order to take calls (I work IT for a hospital and am on-call quite often). But I also receive pages from the hospital in text form, which is problematic if I'm not hearing them.

Is what I am asking for even possible as it stands right now? If it isn't, how would I go about making this a suggestion to the iOS developers?

Thanks so much for any assistance y'all can give.