Please mention whether you’re on stable, beta, or a different update channel. There’s a good chance most bugs on the stable update channel have been fixed already, so anyone on the beta update can let people know if the bug they’re facing is already fixed.
This is a Bluetooth issue, not Linux or Deck specific. When microphone is enabled on a headset, Bluetooth will switch to a different audio codec. That codec supports one output channel (microphone) and one input channel, so it will be mono audio. The sound quality is good enough for audio calls but not for anything else. The only thing you can do is to disable the microphone. Or, you can do what people have done for couple decades now: complain at Bluetooth Special Interest Group until they improve the standard.
Yes it is not Deck specific, but Linux specific. It works on Windows / Android /IOS. And contrary to any other Linux installation, the deck is a commercial product. As a commercial product I want modern Headsets to work properly as they do on other systems. Proper Audio I/O is necessary for any multiplayer game.