I also descend from William Anderson and Bridget via their son William b 1833 and his daughter Caroline b 1857. Caroline's 13th child Alice Anderson is my great grandmother. It sounds like you have much of the info that I have however. I too have searched and come up bare for records relating to her birth in Ireland.
I have been focusing of late on Bridget's last husband George Edward Denham who died in 1903 in Port Macquarie. I have been wondering if her second husband James Edgar and George are one and the same person and there is no marriage record for George and Bridget and also interestingly no death record for James Edgar...
I too live in Port Macquarie and so have access to the library here where the local paper has many entries pertaining to this well known family of the district.
I would be more than happy to swap info and see if there is anything either of us has that the other does not. My email is firstname.lastname@example.org
Looking forward to your reply