No it wont. The character still has the same hit\hurtboxes. But they will not be able to see the texture hack(obviously since they don't have it).
That's the obvious though.
Anything imported to a character (Rosalina holding the hand/Ganondorf having a sword) will cause Wi-Fi to desynch. Heavy vertex editing and adding new textures to the model will also probably cause it to desynch.
I just found out that mariokart64n's Mewtwo model desynched because of he probably added textures to the tube thing, and if the other player's Brawl data can't read such thing on his screen, it will desynch.
When I look at that Mario image, I don't even know where that wing thing came from to his hat so it's not a vertex hack. Since it's probably an import, it might desynch.