Why not? I've done it. All that's required is to remap the pinouts of the logicboard connectors, an easy thing to do with those molex connectors. There's a fair amount of voltage adjustment to follow if you're going to use it that way and you'll no longer have software control over brightness, but everything else will work. The Classic IS an SE. Apple hardly re-invented the wheel there, right down to the ROM.
After reading this repair tour-de-force, I would much rather tackle the logic board swap! LOL