Current workflow is to manually name the Cut List folders AND individual parts in SW. Then, in Mastercam use the auto part names feature in Geometry Nesting.
SW refs only the Cut List folder names in BOMs (check 'detailed cut lists') but Mastercam will only ref the cut list names.
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(IMHO, Detailed BOMs work better than weldment cut lists as it parses the list down to only the parts shown in the selected view, our multibody parts have a lot of 'multi'.)
But really, getting the names onto the parts in the most efficient way possible is the goal.