That is weird and a bit disheartening. How does Wii compatibility mode even work? Can you play Wii games on the gamepad? (Assuming not.) But you can play Wii VC on it? And you can't go into Wii compatibility mode without the aid of a turned-on TV. Nintendo does such weird stuff. Reminds me of how the 3DS has to go into a compatibility mode (and shut off many of its features) to play DS games--and can't even go into sleep mode when playing GBA!
Only wii U titles and wii u eshop games work on the gamepad. There is a wii emulator app on the start page of the wii u menu which gets you into the wii mode compatability and you can do all your wii related things there (including playing/redeeming/buying wii vc games).