You must use a Gamecube controller to play Gamecube games on the Wii. You also need a Gamecube memory card if you want to save progress in Gamecube games.
The Wii basically runs natively as a GC when a GC game is launched (it is not emulated). Therefore there is no bluetooth connection to the Wiimotes (which is why the Wiimote won't stay connected or re-connect), it only polls the GC controller ports. This is also why a GC memory card is needed for saving since the GC side of the console has not been reprogrammed to access on board memory, it only accesses the GC memory slots.