The story: Mary Isabel was born 26 Apr 1819 (or just possibly 1818) in, precisely.... Scotland

Her father was Peter Giles, an artist who had left his wife Jean Hector and his two children (one of whom the artist James Giles, RSA) in Aberdeen around 1814-15 and supposedly went to Glasgow before eventually settling in Belfast. Mary Isabel was supposed to have been named after her mother, who died at her birth or shortly afterward. She was brought up by her father in Belfast, but all later records state her place of birth as Scotland.
So... how does one find a Mary Isabel giving birth to another Mary Isabel on a certain day, without knowing the surname or the exact parish? Or, because of the illegitimacy, is it likely that the birth was never recorded and the needle isn't in the haystack after all?
Thanks for any suggestions,
Sarah