J.M.
Are you staying in a hotel where she'll have the cake? Is she eating it at the competition? I'm still confused. If you want her to have cake on her birthday, then I'd recommend doing so in private either later that night or whenever you have time away from others since it's not a birthday party. I guess the way I'm reading this is that you're going to a competition where other people will know it's her first birthday. Is the competition at someone's house? A gym? A dance studio?
Without knowing more, I say do the cake in private and don't worry about other people. If they bring a gift, maybe save it for her party or open it in private so others don't feel like they should have brought a gift to something that wasn't her birthday party and then follow up with a thank you note.