the main concern here would be the streaming service. If the streaming service offers a full API for custom implementations, then you're probably fine. If, instead, you're using their non-public API, you'll probably get a Cease And Desist and your API keys revoked.

Then, like the other response mentioned, there's the streaming aspect. You, probably won't get sued, but anyone that streams with that music will get muted on Twitch and Revenue Claimed on YT.

Why build a minecraft mod for this, when you can just play music on your computer and turn off the minecraft music? A few hotkeys and you don't even have to swap screens to change the music.

Music in games can be contextual, and it's more fun if you score that beforehand than stepping out and doing it as you go. You may also want to enjoy the score a map/modpack maker intended.

Since that’s all client side, I don’t think sync rights come into it… unless you start streaming while playing the mod, in which case you have now synchronized the music to video and are in violation.