As far as I know, the GameCube controller Adapter only works for Nintendont (ya know, that GameCube games loader) as it supports HID controllers... I don't know if it also works with some other stuff such as emulators for the Wii as well.
I think there is a way to get this adapter to work on Wii games on vWii (as long as you have either an original adapter from Nintendo or a third-party one from Mayflash as these are the only ones who actually work), but only with original game discs, and not with USB Loader GX, sadly...
I may be wrong... But I think your best option is to play it on a Wii (if you have one with backwards compatibility), or getting a HORI Battle Pad, which is a GameCube-like Classic Controller you can plug in your Wiimote...