Best Buy: Intec GameCube Battery Pak (Black) Black G5276
Amazon.com: Hisewen Wii U Gamepad Battery, 2200mAh Replacement Rechargeable Battery Pack for Nintendo Wii U Gamepad WUP-012, WUP-010 : Video Games
SuperMegaPodcast — 4Gamers Gamecube Battery
Closed the case! And it works, fully charged and been running it for 2 hours so far and more to go. From outside you wouldn't know it's now full of Lipos :) :
Buy GameCube GameCube Rechargeable Battery Pak (Black) by Intec | eStarland.com