That's an interesting direction to go with. I take it there is no wall switch to control the light socket in question?
My shave den has two overhead fixtures, one bulb each. One is directly over the sink, the other is also the exhaust fan, pretty much in the middle of the ceiling. It is, for the most part, adequate. That is, until I shave my neck & under the jaw. Then, everything is in a shadow. I added a mirror that extends from the wall, which helps my shave, but not the lighting.
My contribution to the thread, a lighting option I've considered:
https://www.amazon.com/gp/aw/d/B00HS..._t2_B00IZA2URS
You have to buy the power supply separately. I'd mount the strip of lights along the 3/4" wide top of the backsplash of the bathroom countertop. The idea being that it would shine light upward, eliminating the shadow. I tried brighter (and cooler/daylight color spectrum) bulbs, but that did nothing to improve the lack of light under my jaw. In fact, it kinda "washed out" the colors, making it sometimes difficult to see my not-so-thick stubble :shrug:.