The Library of Birmingham does have some log books/registers for Birmingham schools and as you say you can only access records more than a 100 years old [but up to 1925/26 now]. That said, the collection is very hit and miss but you may be lucky.
The Kings Norton School Board covered a much wider area than just Kings Norton. For example it set up Clifton Road schools and Tindal Street school. In about 1903 responsibility was transferred to the new Kings Norton & Northfield Council which continued until 1912 when Birmingham took over.
I wonder if Celia attended one of the Kings Norton Board Schools rather than a school in Kings Norton itself? May be difficult to check without knowing the name of the school. May be worth checking the Allcock School records again - just a chance that it named the school Celia transferred from?