As I wrote the report for my client, I was thinking that this would make a good case study for publication - there was plenty of conflicting evidence, and no real direct evidence linking Katherine to her parents or siblings. And just because I want to know everything about everyone, I started doing descendant research on the Clooney children. The oldest, John Cluny, married Mary McCarthy about 1882, and they had six children: Mary, Margaret, James, Catherine, Helen, and John. Anna Cluny died unmarried in 1892. Dennis Cluny married Mary Skelly in 1887, and they had Helen (b.1888), Margaret (b. 1891) and James (b.1897). There were several names in common with Katherine's children (Margaret, John, Helen...) but perhaps not surprising in an Irish Catholic family.
Then I got an email from my client. He had had DNA testing done with both Ancestry and FamilyTreeDNA, and gave me the username and password so I could go take a look. And lo and behold, one of his top matches in Ancestry had this family tree:
Much more research lies ahead, but DNA has provided the direct evidence I needed to let me know I'm on the right track.