There was a post not too long ago (I think within a year) from someone who tried to bring a cake into Chef Mickey's. While they were allowed to bring the cake in, they had to keep it at their table, and the restaurant would not provide utensils and plates to serve it (plates and forks could be gotten off the buffet, but what's problematic is a proper knife for slicing the cake because the restaurant wouldn't provide one).
The poster had wanted the restaurant to keep the cake in the kitchen, bring it out and serve it as a surprise for the birthday person, and the restaurant refused. They cannot keep outside food in the kitchen or serve it to guests. The restaurant would not even supply birthday candles so be advised you would have to bring those as well.