In 1702, Christopher, Jr. did get rid of the French in St. Kitts: http://caribbean-guide.info/past.and.present/history/spanish.succession/
but I find them mostly on Barbados. Have you checked in England? Walne's 1973 A Guide to Manuscript Sources for the History of Latin America and the Caribbean in the British Isles
has an entire entry for Sir Christopher Codrington, Dodington Park, Chipping Sodbury and lists:
title deeds to Codrington family estates in Antigua and Barbuda, 1696-1837; Plantation accounts, c. 1750-1912; Estate management papers, Antigua and Barbuda 1681-1863; Bills of lading for produce shipped from Antigua and Barbuda, c. 1715-20; Letter books concerning family and estate matters, 1720-1, 1751-61, 1783-91, 1853-5; Maps, Antigua and Barbuda estates, 1750-1800.
I haven't found any reference to Codrington in St. Kitts.
Cardin - St. Kitts