If that label is a decal, and I suspect it is, it wouldn't even show up in a SolidWorks drawing unless it was a shaded view, so I'm not surprised it isn't showing up when you save to AutoCAD. Have you tried creating a SW Drawing, with a shaded drawing view showing the decal, and then save that as .dwg? I still don't have high hopes, but it's worth trying.
Russell,
Simply create your drawing and when you save it select save as and scroll down to .DWG.