It is also possible that the birth mother's was indeed Margaret Davies, and it was incorrectly recorded as her daughter, Catherine. It does happen. Margaret was only 42 at the time of Dorothy's birth. Something as simple as Catherine reporting the birth on behalf of her mother could cause such an error. Sadly, it is also possible that William was both Dorothy's father, and her grandfather.
Research notes are a great place to record cited discrepancies.
I'm not disagreeing at all with your interpretation, or how you handled the profile. It is entirely possible. You have included and cited all the information you have. With what you have, it's either a family secret, or an error.