The 1925 reference for 'Killarney' suggests to me that the Bishop’s Lodge in Hay, NSW, may have corresponded with the Bishop's Lodge in Lucknow, India. The recommended roses in this article from India may be of some value in determining the Bishop's Lodge roses in Hay.
I feel that "Bishop's Lodge Roset's Rose" may well be 'Killarney' which was one of the recommended roses in the article.