I recently ran across similar issues while developing a web app, https://rap.coach , that features simultaneous audio playback and recording. A lot of people are saying that Bluetooth headsets have to go into a super-low quality mode when both playback and recording are enabled, but based on my testing it appears to be a software issue.

The issue appears when I use Rap Coach with Chrome on Android, but doesn't happen in Firefox on Android. It is very frustrating, getting better control over the audio hardware is one of the main reasons I'm going to end up building app store versions of this thing.