I'd change Customs.
First; a Custom shop, the purchase of custom equipment and moves would be available here. Equipment changes throughout the day, while moves can be purchased for each character at a player's leisure (The possibility of obtaining moves and equipment would remain in Classic, All-Star and Master/Crazy Orders).
Second; add custom move selection to the character select screen. After selecting a character (When customs are on, of course) the player gets to select their moveset(s) without needing to enter an entire other menu just to assign a moveset (This could eliminate the problem of restricted movesets in custom tourneys).
Third; eliminate/replace some of the more lackluster moves (I'm lookin' at YOU, Effortless Blade!).
Other than that, I'd try making Heavyweight character's attack more capable of breaking shields; most players will shield a heavy's attack, and then retaliate while they're stuck in atrocious end lag. Perhaps increasing shield damage would encourage most cautious approaches to heavyweights.