Would this be the Stephen Dunphy you are referring to?
Administrative and biographical history: Dunphy and Sons was a company owning grocery shops, bakeries and wine and spirit retails in Llandudno , Craig y Don, Conwy, Deganwy and Penmaenmawr (Caernarfonshire). The company was formed by Mrs Dunphy, who was succeeded in 1857 by her son, Stephen Dunphy, grocer, from Temple House, Mostyn Street, Llandudno. His wife was Catherine, nee Phillys, of Penrhyn Old Hall. They had four sons, Bernard, Richard, Arthur, and Jack. Bernard Dunphy joined the navy, serving on the HMS Conway 1897-1899, and died in 1915, on the auxiliary cruiser, Bayano. Richard and Arthur went into the grocery trade with their father. Jack became a vet and also fought in the Boer War in South Africa. Norah Dunphy, daughter of Richard Dunphy, worked for Tynemouth and North Shields Corporation as an architect. The business closed in 1972
The only Catholic Church in the town that I know of is "Our Lady Star Of The Sea Roman Catholic Church, Llandudno" but someone who is a Catholic may know the one you have named
I found out this info about the Church above:-
The Roman Catholic church of Our Lady Star of the Sea (Ein Harglwyddes Seren y Môr)
stands at the junction of Lloyd Street and St. Mary's Road and was built by J.D. Williams
of Knighton in Radnorshire to the designs of Edmond Kirby, architect of Liverpool.
The church was dedicated on August 17th 1893 by Most Rev. Archbishop Scarisbrick, OSB,
assisted by the Bishop of Northampton, and the Bishop of Shrewsbury
in the presence of His Eminence Cardinal Vaughan, Archbishop of Westminster.