I would expect BMI, ASCAP, or SESAC to cover that, and I would expect it to be very expensive. You probably would need to avoid showing any lyrics or screen images on the stream.

I have been down this road and can tell you that Congress has granted a limited exception to laws stopping a FCC controlled radio station from broadcasting recorded content because of freedom of speech. Ads are one of the limits. The Performing Rights Organizations are in direct opposition to the exception but have been overruled in-part by the Supreme Court. Live performances still have licensing rights but have very strictly controlled rules governing those rights.

Seems the main problem here is that Congress gave the FCC terrestrial radio control which does not apply to non-over-the-air broadcast rights such as the internet. The FCC however has assumed these rights but because of its uncertainty over internet control has declined to fight internet battles yet. However waivers of payment and ownership should be required of karaoke singers broadcast since their performance is live!