One other suggestion might be to set up a system that awards points to your repeat customers, and to only allow the points to be awarded to those who order an equal amount of light and dark meat thereby incentivising an equitable use of each. You could still offer white only or black only options, but only if your customers agreed to forgo the points.
One problem with this type of system, though, is the tendency for quite a number of people to simply miss the points altogether.
A customer ordered lobster and the waiter brought him one with only one claw. When he complained, the waiter said: "It's been in a fight, sir!"
So the customer said: "In that case, bring me the winner".