It is true, it is the only first birthday your daughter will have. I don't see what that has to do with your eating cake. Your child will be oblivious to whether you eat the cake or not. Cake eating is not a testimonial to celebrating and rejoicing on your baby's special day. You could run the video cam or keep yourself busy during cake cutting/eating time in many ways. You could make yourself a LC cheesecake and eat that when everyone else eats cake if it's an emotional issue and you do not want to be left out.

Birthdays are all about memories. Memories from the past. Events from the present all too soon turns into the past. When looking back on your child's first birthday, what is it you wish to remember? The cake, or how it felt to not eat the cake? Only you can make that call. Still, in the greater plan, it's just a piece of cake. If it means much to you, then I can think of no better reason for eating it.