I can ask about the school records if you like. Not something I've ever consulted.
Normally, baptism happens 10 days or so after birth (got to give the mother a few days to recover!), so I would tentatively agree with you that birth date is more likely to be December 1877 rather than 1874.
This would make her only 3 years and a few months old when the 1881 census was taken, and 13 in 1891 etc.