I am no expert, but something I have learned is that some lo can tolerate or not really show reactions when eggs are cooked in something like a cake. My Dr. said a scrambled egg may have a different effect depending on the severity of the allergy.
If you have any desire to wait a bit longer, consider baking some egg free cupcakes and serving that to your lo for the bday. That is what we did. EVERY Dr. will give you different advice.
We have family members who have food allergies and this is something I have learned. Hopefully all the real experts will be able to respond further.
Enjoy the bday!!