Running 120V is right out
First, your instincts about not running 120V mains power are correct. For instance, extension cords and AC power cords are made out of a material called cordage. It's made to flex and handle underfoot abuse, but it's illegal to use it as a substitute for the permanent wiring of a building. So even if you could snake the power cord through voids in the cabinet, it's illegal to do so.
In order to bring 120V up there, you would need to search for a damage-resistant wiring method such as armored cable, EMT, Wiremold or FMC that would fit in the space available. Then you'd need to put an extension on an existing outlet (e.g. switch or sometimes, lamp) to allow you to side-exit that box with the armored cable or conduit. Then bring that into a surface mount junction box up there. If this sounds like a bugbear of a job... yeah, you might as well install fish Romex into the walls at that point.
Low voltage rules are much gentler. Which voltage doesn't matter.
Which voltage below about 30V, that is.
So the "perilous journey" should be made with low voltage wiring not with 120V AC wiring. So the easier and "more legit" way to handle this would be to let the light's power block make 19V.
Essentially, extend the low-voltage side of the cord. If you need to buy inline barrel connector sockets and plugs,
Radio Shack Mouser Electronics or Digi-key should be able to provide.