The lack of a known Catherine in James Francis' family and the very specific date for Jessie Sutherland's birth not coinciding with the Jane of John Sutherland(Galway Militia)'s family cast definite doubt on my proposed solution.Your arguments re the names Francis,James William etc are obviously valid but equally don't have birth/baptismal records to support it under the names Burke,Caulfeild or Charlemont.
It is theoretically possible,but highly unlikely,that both strands are true,each yielding a Lady Ann(e) from the different sides of the marriage,allowing James Francis to reflect his heritage in naming the children.It might favour my solution for Lady Ann(e),with the caveats above,as this comes through the female side,in line with family legend.
Let's hope the nudging works and the portrait reveals some new evidence!
Regards
Roger