"Gerald Ross" (clip) I turned a round piece to fit inside the jaws,
drilled a hole in the center then cut it into 4 pieces. An added touch is
magnets embedded in them to make them stay in place.
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Try holding the dowel in the wooden adaptor/bushing without cutting it in
quarters. The wood may be compressible enough to grip the dowel just from
the force of the jaws of the chuck. If that doesn't work, try making the
saw cuts partially through, but leaving the pieces connected at the back.
Then if the jaws press on it mostly at the front, it should close enough to
grip, and stay in place in the chuck when the jaws are open. In order to be
sure the jaws apply pressure only where needed, you could turn the back of
the wooden spacer/adapter a little smaller.

Let us know how you finally solve it.