As you likely know the baptismal records for Killererin RC parish start in 1870. I did search the subscription site Roots Ireland in case Patrick Mulry was baptized in Tuam. The only record I found was a Patrick Mulry baptized April 12 1798 at Tuam with father Dermot Mulry and mother Honor Ward. I did not search for Mary Mahon because the Athenry records start late.
I also searched the Tithe listings for Killererin civil parish but no Mulry records.
Have you considered DNA testing?