I don't know if the system would try to prevent you. But I doubt it. If you have money to play with, you could experiment with booking one trip and try booking another that conflicts with the first. You would only need to pay for the first one. And don't use Whitney, because those include the $15 per hiker Whitney Zone fee.
As for booking the day before... If you are watching the several nights before the actual, you should be able to get a sense of the demand pressure, and base your decision on that.
I wouldn't book two trips -- with your group size, you are talking $120 per trip!
