Hi Elise,
Happy Birthday to your daughter, it sounds as though she knows exactly what she wants on her Birthday and YES you can book before your trip is fully paid for.
Cinderella's Royal Table in the Magic Kingdom is the hottest restaurant in Disney and is the hardest to get a reservation at. For the best chance of booking you must phone at exactly 90 days before the date (Guests staying at Disney World Resort hotels can phone 90 days before their arrival and book the first 10 days of their dining reservations) You need to phone at 7am EST and my advice would be to be as flexible as possible, for example say what date you want but that you will take any time between 1 and 2 pm. The number for you to call is 407-939-3463 and to work out when your trip is 90 days away try the calculator at
www.pscalculator.net If you've already missed your date call and ask for any time available and keep trying!
If you can't get a reservation Akershus in Epcot is a great alternative, lots of the same Princesses are there and it's a lot easier to get a reservation for.
As you're at Disney for your daughters actual Birthday make sure to sign her up
HERE for a treat during the year of "What Will You Celebrate?"
Good luck,
Jo