Interestingly, a Gamecube adapter already exists granted it is made by a third party company. I happen to own a couple of them before I found out Nintendo was going to provide their own adapters.

Basically, it is an adapter for the Wii remote made by Mayflash, and it makes the Gamecube controller function similar to the Wii classic controller as I was able to move the cursor with it on the Wii menu. I tested this on Brawl and it definitely works fine. As an extra, it can also map turbo and auto fire onto any of the Gamecube buttons. The only downsides I can see with this is probably having the Wii remote die out in the middle of a match or if Wii remotes are not compatible with the new Smash game. It will probably still be a a better controller than the classic controller if used for other games that are not originally compatible with the Gamecube controller.