They have a time in the show in which they recognize anyone that is celebrating something - it's really fun and cute. They don't do it like a regular restaurant would, it's part of the show.
I know people have called about ordering a cake there before and they've been discouraged as the show timing doesn't allow for serving one easily. I'm sure you can still do it, just know it'll be awkward to work into your time there. The only time i remember the show "stopping" for any amount of time is when the food is first served. They even perform during the arrival of dessert - it's part of the show as well. Once the show is over, you're ushered out fairly quickly so that they can clean up for the next show.
I think HDDR is a great place to celebrate a birthday but in a different way from other restaurants - it's a dinner show so it's not easy to have a personal party during it, make sense? If you want to do a big family time with cake and the birthday song, maybe you could do a big breakfast or lunch?
I bet he'll have a blast regardless - we love HDDR and it'll make him feel VERY special!