I finally got around to playing with some 3M matte black vinyl wrap. It was very easy to work with -- thinner than electrical tape. At first, I just cut some strips and went around the bezel in four sections. I was surprised how good it looked and left it that way for a bit. I even covered the trim in the middle of the vent lever. I was curious if I could do it in one piece, so with a bit of heat from a hair dryer, the strips came right off without leaving any residue.
I overlapped the entire vent with one piece of vinyl and worked it around the edges of the bright-work with my fingernail (and a little heat from a hair dryer). Came out pretty well for my first time using this stuff. It's not perfect, but I needed tinier fingers to work the corners better. Good enough for 15-min of work... and I like that it's reversible.
Annoying window reflections is GONE!