Just make sure that Twitch or Mixer does not re-broadcast the music stream. Twitch could decide to mute the twitch audio or some label/publisher might file a claim.
The GridStream Webplayer should have no issues running "in the background" so people can have that and twitch in two simultaneous browser tabs for example. Or use the desktop GridStream Player.
The webplayer can be found by going to
http://player.gridstream.org/Download urls and other info can also be found there. This is currently one of few legal ways to tune in.