How do I select audio source when connecting iPhone to HDMI monitor?

I have a USB-C HDMI adapter connected to my iPhone 16 Pro Max and the other end into an HD monitor and it mirrors the picture.

I set the audio on the iPhone to connect to bluetooth speakers, but the sound comes out of the monitor speaker, which is not great.

Does anyone know how to do the settings so audio goes to the bluetooth speakers?

I have tried to adjust the monitor audio settings to select an audio source, but nothing happens when I click that button. When the HDMI contraption is not connected, that audio select shows up on the screen and works fine.

Also if I set the iPhone bluetooth settings to Airpods Pro 3, the sound does go there instead of the monitor. It would seem the speakers should work in the same way.

Thank you for any advice.