Iphone 13 mini bluetooth notifications are silent

I'm on ios 5.1 now with no progress to this issue. As I have written in my original post, I have tried it with a really old simple and small Philips speaker, and this problem doesn't happen. However I have also tried many other phones with my Bose speaker and there's no problem. I just bought my iphone 13 mini and there was no BT text notification from the start. Another thing is when im on BT but there's no music on, the call ringer comes from both speaker and phone but with half volume which is also very weird, as if its dividing the notification sound btw my phone and speaker. I'm sure my speaker is perfectly fine. Is there no solution to this?

I have already tried these, along with many other settings. Nothing works. And there's no such options in bluetooth settings neither. In addition to that, I think you have misunderstood the problem. The problem is that, when I'm connected to my Bose soundlink mini 2 bluetooth speaker and have music on, my text notifications should be coming from the speaker and not the phone. Phone should pause the music and notify me through the speaker when the message arrives. Instead, the sound comes from phone and with half the volume. Looks like a bug to me. I'm installing ios 5.1 now to see if it helps.