When you check in for your ADR on the app, it gives you the option checking a box for different celebrations.... birthday, anniversary, etc.. However, there is a little note underneath that says something about "no complimentary dessert". We were there for my son's birthday two weeks ago and despite checking the 'celebrating a birthday' box on the app for multiple different reservations, none of our servers acknowledged it. I asked if they could do a candle or anything on our dessert at Liberty Tree Tavern, and that was a no.
We did end up getting the Mickey Celebration Cake at the pool that night from CBR and it was delicious. If you end up going that route it would probably be just right for 4 people. If not, there are special cupcakes at Main Street Confectionary in MK or you can mobile order at All-Star Music. If you log in to the app you can pretend to do a mobile order and then cancel if you want to see what's available. It looks like they have a Cookies 'n' Cream Sundae Cupcake for $5.79 available now that's very cute!