I've been mixing from the stage for more years than I'd like to admit. In this case, I'd use an Aux mix with all vocal mics set at the same level, and put nothing else in the wedges.
I think the confounding factor here is that a singer already hears their own voice through their head (perhaps via bone conduction). That seems to make them need their voice loud enough to overcome that effect.
Assuming all heads are equal, I think that would not be a confounding factor.
In the my 2 cents category - if I were the 3rd best in the world at anything, I should be able to afford a competent balance engineer.
