I'd also recommend Inventions. I would call Café Mickey overcrowded and average long before I would call it lively. There's much more variety of food in Inventions plus the 1-year old will be free there – at Café Mickey she would be expected to share from an adult's plate or a kid's meal would need to be ordered.
You can order a birthday cake if you wish – preferably when you reserve your table but at the latest upon arrival. Some considerations about this: it serves 6-8 people so you will have a fair bit left over (the cast members will be all too happy for you to take it away in a box if feasible but sometimes it is an ice cream cake so you would be wasting half of it), and they will play music over the restaurant's speakers and have several CMs and characters bring the cake at once, which may be overwhelming for her.