The solution is the same one used by airlines and hotels: dynamic demand-driven pricing, charging enough that you don't sell all your seats until the last minute.
An airline that sold out of all its seats 6 months ahead of time would be considered incompetent, but governments do this all the time with reservations for various activities, because they are trying to "be nice" by charging low entry fees.
An airline that sold out of all its seats 6 months ahead of time would be considered incompetent, but governments do this all the time with reservations for various activities, because they are trying to "be nice" by charging low entry fees.