Make Dark Link a manifestation of accepting mundane existence. Part of childhood is thinking the world is magical, special, and wondrous, but as we grow older, the burdens of life begin to weigh us down, and we give up childish dreams and idle fancies.
Turn Dark Link into the fae version of Paradox for mages; he's the lingering doubt that none of this is real, that Link isn't really special, that this is a fool's errand, not a quest. He's the whisper in the back of your head that Zelda is just a girl, that Link's tools are just a bunch of trash and rubbish, that it is all pretend. It's time to give up wandering around, there are real world things to do, work and obligations and responsibilities to take care of. Come on, time to let go of these fanciful whimsies. Time to grow up.
The sword, the hookshot, the boomerang, none of these work against Dark Link, they turn into trinkets when they touch him. The only way to defeat him is to find something so special, so amazing that it dispels all doubt, restoring faith in the Quest.