Stream/Striim sounds fine.
The spelling looks weird, but the way it sounds and flows with the naming of games sounds much better than Current. The spelling would just be to relate it to Wii in some way and easy trademarking. adding a small x in front of it for game naming makes everything sound EXTREME though, and that is already better than Current.
Mario Kart xStriim, 1080 xStriim, xStriim Sports
The marketing is too easy though and would likely get abused, but still sounds better than Current.