Yes, the Babycakes options at BC are at Beach Club Marketplace.
Yes, they do have 6" cakes that you can order for a birthday. You want to contact them ahead of time because they only make them to order and don't have them sitting around just in case somebody comes in asking for one. The cakes can not be paid for using the dining plan and are not cheap (last year they were $45+tax if I remember correctly). Here's the 6" cake we got last year. They have lots of options for cake types, icing flavours and icing colours as well as decoration styles. They're great to work with.