Likely what is happening is that the game is probing audio devices and triggering the mic on your headphones to get picked up. This switches them into the "headset" profile which has awful audio quality. I don't know why the UI isn't showing that, make sure you are checking while the game is running and the audio sounds bad.
If you want your headphone mic to work there is not much choice. There isn't a standard bluetooth profile with good audio and mic. If you never want to use your headphone mic you can probably configure some advanced settings in your audio manager (probably PulseAudio or PipeWire).