if you use a Wii DVD drive on a GC you will need a modchip no matter what to play your backup copies. the type of modchip will depend on the DVD drive...
the most simple solution will be to just use a modchip like WKF/WASP to play your copies from SD card rather than carrying around DVD discs...plus you will gain usage / battery life since the WKF/WASP use considerably less power draw than a DVD drive
is there a specific reason you want to use a Wii DVD drive?