To you & the PP’s: stop assuming that OP’s DD (or anyone) has friends, and friends who would “take them out” at that, and stop assuming that OP’s child has any interest in drinking.
OP said that her DD is anxious about having sufficient time to see her friends before she moves - this suggests that she has friends that she will want to see when she's home. She may or may not want to see them all at once at a belated birthday event.
Assuming that she has friends in her study abroad program, she will have the opportunity to celebrate her birthday with friends and may not feel the need to do that again at a later date. I would ask her to save time for a nice family dinner and not monopolize her minimal time unless she wants you to plan a celebration.