This is really bizarre. I take it there isn't on-site supervision of the cemetery.
You'd think someone would have had to provide authorization to remove stones and install new ones.
Have you contacted the Beaufort County Historical Society, http://beaufortcountyhistoricalsociety.org/
and the Beaufort County South Carolina Genealogical Society, www.beaufort.scgen.org/
You mentioned that you contacted the monument companies but if there isn't anyone on-site at the cemetery, you may need to cast a wider net. I think I'd ask one of the local companies who do they contact to get permission to install a monument in the cemetery.
I'm not too surprized by the city's response. They probably have the record (unless they purge them) but don't know how to find it with just the Worthington and Ward surnames. Hope you find an answer.