No. Let me make this as simple as possible. I am listening to music while playing a PC videogame. I am also chatting on my mic while playing a PC videogame. I am also listening to my PC videogame. What I want to do is allow the other people I am playing with to be able to hear my music, even at the expense of me not being able to hear the game.

There's no way to do that other than just putting your mic up against the speaker. And I'm hoping whoever you're playing with actually wants to hear your music and you're not just trying to be annoying like people tend to do.