Re: If Anne had a daughter her parents names would be there
somewhere. If her mother name was Bridget then likely that is who she would be named after. In those days most names had family connections as rarely were other names used until the 1900's where you might find people using a name because they liked it not because it was a parents, uncle and aunts, grandparents, siblings name.
www.familysearch.orgBoth have RC collection included in the record:
Martinus Hart born 30 Oct 1845, Drumcliff & Magherow, Sligo
Fa: Michael Hart and Brigida Banks
Anna Hart 27 Dec 1850 (I'm surprised there isn't another child between them)
Annette Code