They sent out an email right before launch that says it's not recommended to have any other streaming service running while you're playing. Past that, I don't believe there is currently a way.
Chromecast can only stream one thing at a time, so you can't use it to cast music AND Stadia simultaneously.
Was doing it with next to no hassle with mi box s, using an ethernet adapter/usb hub and the apk from playstore copied over, was using the PS4 pad too via Bluetooth. Just got an official stadia pad and ultracast and it works so much better visually