I appear to be a year late with a potential answer. Nevertheless, here is the solution that worked for me in my particular case in both Windows 10 & 11. This works with SWTOR graphics settings set to Full Screen (Windowed).
In Windows 10 go to Settings > Sound > Sound Control Panel. Choose your sound device then click Properties. Next click the Advanced tab. Under Exclusive Mode uncheck the "Allow applications to take exclusive control of this device". Click Apply, OK then exit.
You can now click on the other monitor and the music/audio will not shut off but continue to play in the background.
I hope this fix helps you or anyone else experiencing the same issue.
Sadly this doesn't work for me. I tried everything already. I can't get it fixed since I changed monitors a month ago.
