There was a post awhile back that went something like this:
Poster wanted a special cake for a birthday child. Decided to save some money by purchasing the cake at the Boardwalk Bakery, taking it to Chef Mickey's and asking the restaurant to hold the cake in the kitchen, then bring it out with lit candles, present it to the birthday child, and serve the cake.
What ended up happening was: she was allowed to bring the cake into the restaurant. But they would not allow it to be kept in the kitchen; she had to keep it at the table. The restaurant would not provide staff to present or serve the cake, or plates/utensils to cut and serve the cake (these could be gotten from the buffet); nor would they supply birthday candles. Even though the poster had purchased the cake at WDW.
So the answer is quite likely that you can bring the cake as long as you are prepared to keep it at the table, cut and serve it yourself; and bring your own candles and probably something to light them with.