Even 'listed' lighting system Puck lights will have the same issue - they are listed as "portable furnature" lighting or something to that effect and the listing also prohibits them being installed behind walls..... And as a system you can not modify the components.
LED pucks.... Be prepared to spend!!!!! (for anything decently made) And oh yes - same code/listing issues - but less heat.... Dimmable$$$$ And the light quality is in the toilet.....
At any rate - be friendly with the cab guy, and charge more for coordination - he need to build false backs and route out false shelves for you to install the stuff in. (Have you ever wired a guitar?????)
see attached: I did this with LED pucks, and a small LV relay with alarm reed switches. But the basic concept is the same.... Sandwich the lighting between two cab pieces the normal bottom one gets the hole for the lights, and an additional one just lays on top of it inside the cabinet to contain the wiring for them, and a false back for the trans or driver, or J-box.