Well... It would probably "work", but I wouldn't recommend mechanical relays in this application.
They are basically a mechanical switch controlled by an electromagnet and they don't "like" to be switched on & off many times a second.
And, they will "click" or "chatter", which could be an issue.... Probably not a big issue if the music is really loud!
There are solid state relays that operate from less than 15mA. I don't know if you'll find one you want to use, but for example this one
has in input resistance of 1000 Ohms, which means 5mA @ 5V. But, it's more bulky and more expensive than the one you are planning on using.