I thought Rootschatters would like an update.
I have been in touch with the Little Sisters of the Assumption, and have just received a lovely detailed reply from them.
This was indeed the lady I was looking for, she was my husbands grandmothers half sister. The reply has confirmed her birth date, and her parents, and details of her religious life. She was one of two sisters sent to New York to start a new house there.
The name she took on clothing was Sister Mary Benigna - the “Bergine” we puzzled over.
Sadly she died at the age of 37, in 1926.