Never had an issue with online when using a Wavebird other than the usual issues that's non-controller related.
As for batteries, a set of aa's will last ages.
With wireless controllers being banned from some tournaments it's for a number of reasons. Probably the biggest is to remove people complaining about interference from wireless devices affecting it. Another reason is that wavebirds, unlike wiimotes, aren't synced, they just have 16 preset channels which means that some random in the crowd might bring theirs and put it on the same channel as you to mess with you mid-match, or alternatively, you could use that person to play the match for you with you pretending to play.