Here are the original proprietors:
Duke of Albemarle, Earl of Clarendon,Baron Berkeley of Stratton, Earl of Craven, Sir George Carteret, Sir William Berkeley, Sir John Colleton, Earl of Shaftesbury.
Which of them set foot on Barbados?
“The Lord Proprietors did not travel to Carolina themselves but they sent their agents from Barbados to oversee the settlement of Carolina.”
https://www.slideshare.net/mobile/capesociology/before-there-was-carolina-there-was-barbados-30167075
You name the “titles” of the Lord Proprietarys. For example, your Earl of Shaftesbury is aka Anthony Ashley Cooper. Couple that with this:
“Of the eight Lords Proprietors to receive a charter to settle the Carolinas, the most active were lords Anthony Ashley Cooper and Sir John Colleton, both of whom owned plantations in Barbados.”
So it would appear that at least two of your Lords, though they may never have set foot in the New World, certainly had ties to Barbados (and then to Charlestowne via Barbados). They sent their agents from Barbados to the Carolinas. Sounds a lot like a tie to me.
FYI, “The titles of the eight initial owners reflect their rise: Edward Hyde, first earl of Clarendon; George Monck, first duke of Albemarle; William Craven, first earl of Craven; Anthony Ashley Cooper, first earl of Shaftesbury; John Berkeley, first baron Berkeley of Stratton, and his brother Sir William Berkeley, governor of Virginia; Sir George Carteret; and Sir John Colleton.
https://www.scencyclopedia.org/sce/entries/lords-proprietors-of-carolina/
The answer to that would be Sir John Colleton.
“Sir John Colleton, 1st Baronet (1608 - 1666) rose to rank in the King's army during the English Civil War, spent £40,000 in the service of King Charles I, and lost much more when his property was seized by the forces of Parliament. He retired to Barbados and was one of about a dozen Stuart followers there who were knighted by King Charles II.”
Even you must now agree that, at the very least, there were ties between S.C. and Barbados! Will you?
https://www.carolana.com/Carolina/Proprietors/johncolleton.html