Blizzard, I just inform you that headsets on wifi/bluetooth mode do not seem to work properly. This bug has been reported several time by the community.
I am using Edifier W820NB and whenever I use them in WiFi mode, the game emits a metal sounds in the background. It is clear to me based on search carried out on Reddit and google that there are various bugs concerning the use of headset in wifi/bluetooth mode.
I wonder if Blizzard Devs could put some efforts now as they are working on a new patch that is coming soon.
If the headset comes with a USB dongle for wireless non bluetooth and you run into issues you may consider consulting with the headset manufacturer for details on troubleshooting this issue. If they report back any useful information, posting that here for others to find is always helpful!
BIG issue is that Starcraft 2 is using microphone INPUT which causes bluetooth headset to switch to headset mode resulting in a low quality output or no output at all (affects all running applications).
Solution is for SC2 to stop using microphone!!
In the past workaround was possible by changing settings in Settings/Voice (had to be done every time the game was launched !!). Now the setting submenu was hidden (resulting in a ugly gap in the menu btw) without actually removing the function, so the workaround is no longer possible.
Not sure if it’s the same issue that OP is having but still …
I personally think the problem is the Starcraft II client. Probably a technical revision of audio by the Blizzard dev team might be good. I’ve been trying different options by none seems to be working. I don’t have such an issue with other games.
I will try again when I feel like it. I lost hope pretty much.