There are many ways to skin cats - - -
I think I would rough-turn the blank between centers, put a tenon on one end, and mount it in the scroll chuck. Turn the body, making sure that the diameter is correct at the point where the body of the turning meets the metal scoop.
Is the shaft threaded? If so, I would drill and tap the hole to match the threading, and then part off leaving the end of the handle as clean as possible. Then, mount a length of threaded rod in the scroll chuck, and screw the handled onto the rod to smooth over and sand the end.
If the shaft isn't threaded (which I presume means that it must be glued to the handle), I would drill a matching hole and part off the end of the handle. At this point, there are two options. You could mount a length of rod or dowel in the chuck, but since the shaft isn't threaded, it won't work well as a jam chuck (although you might be able to wrap a couple of layers of masking tape around the rod to pad out its diameter enough to jam the handle onto the rod. Alternatively, you could make a jam chuck with a slightly oversized tenon (dowel) extension to hold the handle while sanding the end.
Or you could mount a sanding mandrel in your chuck, and hand-hold the handle while sanding the end.
Like I said - many ways to skin cats.