It's as you suspect OP. So many people are used to the GC controller and have overhyped it IMO. The GC controller has several weaknesses in comparison to the the Wii U, biggest being analog triggers, 1 less shoulder button, and a gimp z button. People claim the adapter does not lag and the Wii U controllers don't, or vice versa. Honestly, no one has done a concrete input lag test on either. Personally, I don't see the lag, but again, this is subjective.
The main concrete advantage the GC controller though is logistics. It's easier to setup.
As for the short battery life of the Gamepad, you are right. If you are willing to invest in a high capacity battery pack, the Nintendo store is selling them on their parts website.
https://store.nintendo.com/ng3/browse/productDetailColorSizePicker.jsp?productId=prod150204
I can attest to the extended battery life with just the High Capacity Battery. Some people say this combined with a certain 3rd party battery extender, you could go for an extremely long time, longer than most game sessions. Maybe even longer than a Smash 'til Dawn.
In the end though, I just go with the Wii U Pro controller. (I have the adapter, and don't use it). I use L to jump so c-stick placement doesn't bother me. Notches I don't care for.
Expect many responses of go with whatever feels best. I'd say **** that and go with whatever is most efficient. As of right now, there's no clear cut winner. But for now, I'd say you're fine with the Gamepad.