Don't have any point of reference for input lag but the pro controller is really nice and I personally can't tell that there's any input lag. Being bluetooth there is probably a very small amount but in my experience not enough to be noticeable.
GameCube is the go to, but it really matters what you're most comfortable with.
GameCube has benefits because of the notches on the sticks making directionals easier and more consistent
Pro controllers are a good alternative.
I don't really know about wireless that's really up to you
I use GameCube controller with 8bitdo wireless adapter it works great. It is also compatible with Wii gamepads. I finished smash bros with a PDP GameCube style Wii gamepad. Not as good as the original but very lightweight for long sessions.