I really enjoyed reading about dyeing synthetic fibers. I actually Googled and found even more in detail explanations of the different processes. They all seemed pretty complicated in the long run. You know what I would do? Buy a second wig and cut it up to make the beard.

That way you have an exact match on color. I really is not that hard to do. A wig is made up of lines of wefting in the back that you can cut apart and then sew or glue to the face of the puppet how you want them.