Hey wackyquack,
This is always going to be a problem but to me it's simple. The patrons who are there all night spending their money to keep you in business are the ones to keep happy. If those that come in at the last minute and think they can sing quick and often really want to be treated fairly then they'll come early and stay late. Those are the ones that I will assure that the rotation is full for the evening and to come earlier next time.
The way I generally do the rotation is this. Wherever I am in the rotation at the time they turn in a song request, I will insert them about 3/4's the way down the rotation. e.g. If I have 20 singers and I'm on the 12th singer in the 2nd rotation then they will be inserted about 7th in the next rotation. I will fudge this some with someone that has been there for quite some time but is just now turning in a slip. It's just really for those who just walk in and turn in a slip late in the night.
Hope this helps
