First up, the Wii U Pro Controller uses the same battery as the 3DS. The 3DS battery is named CTR-003. This info comes from the Wii U instruction booklet.
"However, the GamePad and Pro Controller each require specialized lithium-ion batteries — the WUP-012 and CTR-003, respectively — and according to the manual, replacement batteries must be ordered directly from Nintendo."
Look at how much play you can get out of the 3DS battery when it's not in the 3DS! 80 hours of play compared to the 3DS' 5 or so hours of use. I guess those screens do eat up a lot! Thanks to Gil for the heads up.