I would actually caution against birthday cake only because there are a lot of ingredients in birthday cake that Amber hasn't been exposed to yet. They say to try a new food or two every 2-3 days, after making sure that your baby doesn't have any reactions to it. I started my older daughter (no c/f) on cereal and baby food at 6 months, and she actually had some issues around 7 months old after eating green beans. It turned out to be totally unrelated to the green beans, but it was easy to take that one new thing out of her diet until we figured out what was causing the problem.
Cake generally contains dairy, eggs, and a bunch of other delicious ingredients, which will almost surely be fine for Amber, but I would expose her to new things slowly just in case. If there is a reaction at any point, you will be able to finger the culprit pretty easily.
Have a wonderful birthday party!
-- Jenica
Mom to Abby, 2 (w/o CF) and Ellie, 7 months (w/CF)