Well, I don't have actual experience. But we've celebrated numerous birthdays at the parks and the places that choose to give you a free dessert, do so at their discretion. Meaning they are not obligated to give you one and if they do, you don't have to buy it.
Not all park goers are on
DDP, I don't think restaurant management will discriminate
against those on the plan by not offering them a birthday perk.
You pay for the DDP and it stipulates that you get a dessert as part of your Table service, so they would not deny you the chance to choose a dessert if they bring you a cupcake as a celebratory gesture.
On the other hand, if they know it is your birthday, they may ask you to choose your dessert and then put the candle/sing the song to accompany it rather than offering you an
additional dessert.
I think you'll end up with plenty to eat either way!!