Yes, it's done. No idea on the cost, though. Early-mornings are often used for the surfing lessons in the wave pool.
Typhoon Lagoon has nighttime lighting, so it can be rented out after-dark, whereas Blizzard Beach cannot. There's a private event/party that rents out TL every year during Gay Day weekends. (Another group also rents out the Studios) Ticket to those dance party events (which in addition to paying for the park and the lighting and sound equipment rentals also pay for the DJs that play the events, and probably some kind of a profit for the organizers) are in the $75-$100 range.
-Rob