The parish registers for Belmullet can be found here
http://registers.nli.ieThere's an entry around Sep 1854 that reads "6th, The Rev'd Pat'k Malone, P P of Belmullet" which fits the newspaper item.
I have confirmed that conversion includes baptism and have rechecked Belmullet from 1848 - 1856. (began in '48 on basis the elder sisters would not have considered such a step while their father was alive). Cannot spot the name Bingham - though reading the lists is not easy. Suggest you double check and widen the search! The newspaper item refers to 6th Jan (1855)".
If the baptism were recorded would parents' names be included in these circumstances?
Another search option is to look for a Belmullet marriage. Unfortunately, the marriage records begin 1858 - which leaves a gap of 3 years into which a Miss Bingham marriage might easily have fallen.