If you have not succeeded as yet may I make the following suggestions:
1. Since you have your GF's DoB please write the Registrar General's office in the various islands and request his Birth Certificate. On it you will find his mother's name & her maiden name and also his father's name should be there.
2. You can also request a search for DoB and her BC for a child Clementine born in that year with the maiden name found above. You will then get her parents' names. You can continue further on her side that way.
3. Similarly you can try for James Branker's info.

All the best in your research from Shamshu Deen.