The method and results are not perfect, but it cut down on the amount of shine enough that I didn't hate it. The lesson learned is that I need a small makeup kit.
I created a mask around the guys head. Then to the color wheels of Lumetri Color and reduced the Highlights bar to a point I felt was good. I also lowered the luminance in the Curves tab and then only tweaked the Contrast and Highlights in the Basic Correction tab. Then I applied the Red Giant Cosmo effect and tweaked those settings to something I was happy with.
Thanks. I'll give that a shot. You would think by now there would be an effect that's designed to handle this easily. A paint or clone stamp that you can just track through the clip without a headache.