Since I was able to distill the choices down to two, your option is to pay the fee for each rcd and see which one has a mother's name of Bridget. Warning - After Mary, Bridget is probably the next most common female forename. The chance of a false positive is HIGH.
For the sister Bridget, search in the same parish, same county for the birth of a Bridget with this surname in a range of +/- 4 yrs around the date of Patrick's birth. Hopefully you will find only ONE rcd, or at most, a small set of rcds for it. A view of the RIGHT detailed rcd will show the correct father and mother's forenames. This involves a small fee per search.
You really can't write anyone for help. The churches have largely turned over their parish registers to the National Archives for Ireland (NAI), or to the Irish Family History Foundation for transcription. You could commission a search for these ancestors, but the minimum fee starts at abt US$85 per hour and some minimum number of hours. This gets expensive fast.
Genealogy is rarely available for free.
Since you believe Patrick's mother emigrated with him, search the U.S. censuses for the relevant decades. You will likely find her name and age under the rcd for him which lists him as head of household. Also, you can look for immigration & naturalization papers on the two.