I really do think this mode will be wi-fi enabled. You can see the background of a globe, meaning wi-fi is 99% confirmed. Which means, we can do 1vs1 battles in this mode. And if everyone is complaining about double elimination so much, I solved that problem by picking 2 playable characters in the roster, so if I lose once I will have another try, and to fully eliminate my opponents, they would have to be defeated twice.
So this tournament mode would be more for 1vs1 and tournaments. And the other wi-fi option would be for Free for All and playing against friends. And the last wi-fi option, would probably be for team battles and trading stickers, etc, which they havent mentioned yet. I'd be willing to bet that tournament mode will be wi-fi enabled. Sakurai team is probably working on tourny mode to be on wi-fi. He did mention that they are still working on this "seriously hard," most likely for wi-fi, because it looks complete to me, except they are working on it hard to make it playable through wi-fi.