In terms of primary sources here are a few you might try:
The Barbados Archives Department which preserves and protects the records of Barbados and Barbadian life. They have some records from the 1600s, so an email sent to them may help uncover some records relating to Charleston.
This one is more of a stretch, but the Charleston Archive in Charleston, SC, may also have something of use.
The National Archives of the UK has a listing of Empire and Commonwealth records held by other archives that includes a book list that might provide either some secondary sources, or a guide to primary sources. In particular, JS Handler's "Guide to Source Materials for the Study of Barbados History, 1627-1834" might be a good place to check.
Finally, there's an interesting and informative article, "Escaping to Charleston's mother colony, Barbados," in the Charleston City Paper about Charleston's relationship with Barbados that you might check out.
Hope this helps!