I have a three count too. But usually it happens when I have 25-30 singers and nobody tells you they left for the night. In rare cases I could have not been able to find them. Because I never saw them before or they were in the rest rooms. I never ...hardly ever check restrooms. Unless I know that's where they are and it's taking way too long to return.
This isn't going to come out right.
Skipping a person totally seems pretty counter productive to enhancing your reputation. I know there may be some that want to milk there "show" time. To me it shows you may not be as dedicated to your business as you pretend. Like it's a job you're getting tired of.
Don't get me wrong...I have little tolerance for jerks. But this isn't Hollywood or American Idol. It's karaoke at Joes Place. Things aren't going to run perfectly smooth. You are talking about the people that pay your way.
The last thing I want to do is Pi$$ anyone off to the point that they don't want to come back. EXCEPT JERKS...
So you missed your turn...I went to the next singer without missing a beat and you can sing in your new spot in the rotation.
It's not WATERBOARDING...

(I endorse Torture that doesn't permantentaly maims you.)
It's CUSTOMER SERVICE...not fingernail pulling...
Why is it so hard to move them to a different spot??? Without going postal...
You make it sound like it happens all night long instead of once a night. And from my understanding you are "filling" in the dead air anyway.
I also understand you don't want things to get out of hand and become something they think they can do all night, every night...