One possibility for this bone is a phalanx of a primitive mysticete or odontocete. The flattened articular surface is characteristic of Neoceti. Archaeocetes like Eocene basilosauridae (e.g. Dorudon atrox, shown here – from Uhen, 2004) have quite narrow elongate metacarpals and phalanges, unlike extant Neoceti.
This is surprisingly skinny for a neocete, but we’ve got many primitive Neoceti from the Oligocene of Charleston without well-preserved flippers. Most actually lack flippers, so anything is possible.