In my experience, civil birth records in St. Louis are hard to locate until the 1870's. They get a bit better in the 1880's and 1890's, but are still spotty until 1910 when statewide birth registration was required.
For Catholic Baptismal records, my understanding is that the Archdiocesan archives has them on microfilm and has provided a copy to the St. Louis City and County libraries. I'm not sure if FamilySearch has them online.
To search them effectively, you need some idea of where the family lived. Do you know Mary McLaughlin's father's name? Since the 1860 Census does not provide exact addresses, I would suggest looking for him in the City Directory for St. Louis. (Ancestry has a good one online from about 1860 or 1861). That should give you an address where the family lived.
If you can't make it to the library, I would write to the archives. When I was looking for my gg grandfather's baptismal record from 1860, I wrote to the archives, enclosed a small donation (I think $25 or so) and gave them the information I had on the birth date, location the family lived, and parents name. They were very helpful, and searched a couple parishes until they found the record. Their contact information is in the link below:http://archstl.org/archives/page/genealogy