Welcome BACK, Sandra!
How exciting that your family will be celebrating a Birthday at Walt Disney World Resort! Definitely pick up a "Happy Birthday" button from the front desk of your Disney Resort Hotel or at a Guest Relations location. I also think the rest of your party should pick up the "I'm Celebrating" buttons, to really tell the 'World that you're all celebrating something special together!
Domed
Mickey Celebration Cakes are available for delivery at 45+ Table-Service Dining Locations. Since Be Our Guest is technically a Quick-Service Dining Location during breakfast and lunch, I understand why you're being told one cannot be added. While I don't like that answer, I'm not out of ideas.
Talk to the Concierge at your Disney Resort Hotel, and/or the Cast Member at the check-in podium of Be Our Guest to see if they can last-minute add a cake to your desired dining reservation. Each participating dining location keeps a few cakes on hand for last-minute requests, so I'm sure something can be done.
Do you have other dining reservations at another Table-Service meal? Check the list of locations on the cake page to see if one can be delivered there. Are you staying at a Disney Resort Hotel? These cakes can also be delivered to your guest room. That would be a really fun surprise to come home to! The number to set this up is (407) 827-2253.
Lastly,
Amorette's Patisserie in Disney Springs also has adorable dome Mickey, Minnie, Goofy, Donald and Pluto cakes. They have big cakes to feed the entire family, as well as individual sized cakes so everyone can get a different flavor. These can't be delivered to Be Our Guest, but you can certainly bring it with you to the reservation.
I hope this helped, Sandra, and We'll See You Real Soon!
~Kristen