I am hoping for some advice re Oldbury and tracking down the birth reg/baptism of my great x 3 grandma, Harriet Smith. She is born aprox 1848c and in Oldbury according to census returns. On the 1851 census, she is in Oldbury she is with her mother Eliza Bayley formerly Smith nee Hales. She is aged 2.
I am wanting to establish who exactly her father is. She names Eliza's husband Benjamin Bayley/Bailey (married in 1849 in West Brom) as her own father when she marries; however, I believe she was born in the months prior to this marriage and could be a child to Eliza's former husband, Henry Smith. She could even be illitigimate as I cannot locate Henry's death as present due to the high amount of them.
Henry & Eliza are in Oldbury on the 1841 census, and Eliza and Benjamin are still there in 1851.
I have emailed Sandwell register office today but they have responded to say they have no Harriet registered between these years with a mother named Eliza Smith nee Hales. They recomended I contact Dudley office which I have done.
I am wondering if parish records may be something to consider; however, I am based in Manchester so visiting the area isnt an option for me

Any advice is much appreciated.
Claire