There is a birth registered in Galway City in 1865 for a Patrick Gill. (Volume 4, page 314). You might need to check with the local Galway office as to whether its registration area included the Aran Islands in 1865 but I suspect it might. (You can register a birth anywhere now but each office used to have specific geographic areas). If it?s the right birth, you might get a birth certificate after all. You can order a copy from GRO Roscommon. Quote the location, year volume & page number on the application form.
According to this site, it appears that the Aran RC baptism records only go back to 1872, so you may be out of luck looking for a baptismal certificate.