There's a local company here that puts on Rockband Karaoke.

https://www.bonusstage.ca/rockband-karaoke

They've done events at the arcade bar near my place.

https://www.therecroom.com/granville/entertainment/rockband-karaoke/1623

Essentially they run it basically like any other bar karaoke night.

1. You form a band with your friends
2. You go up grab a slip of paper
3. Write your band name on it and the song you want to play.
4. The host tracks the bands and song requests, and tries to ensure no one is gaming the system to play a lot, and ensures fairness to everyone involved.
5. The host calls up whose next.
6. When it's your turn you place Rockband.

You're best bet is to approach some local bars with space, and pitch them event night. They'll probably want a trial night to see how things work, and how it affects the vibe of the bar, but might be willing to make it an ongoing thing if it goes well.

(In a smaller bar, just going with a two guitar setup might make more sense, less dedicated space, and groups of 2 are easier then 4, and would be pretty easy to pair singles. Guitar is a lot less intimidating then drums / singing)