As far as I can see, if custom movesets shouldn't be a huge problem competitively, especially if we compare the options available in this game to the options in other games with competitive scenes like Pokemon, or perhaps a better example, Dark souls and Dark souls 2.
Seeing how the customization in smash seems quite tame in comparison to both of those games, and seeing how both of those games do have thriving competitive communities, I'd say that we would be able to manage, especially once the meta evolved since overtime we would be able to determine which moves are inherently more useful than another move, either overall or in a particular match-up.
If you're plying Marth for instance, and you know that Pikachu's custom 2 variety of thunder jolt is inherently more useful against you in that match-up than the other two, or that mii brawlers do not like to use "head-on assault" in that match then you can make some inferences about the character before they even start using any moves. Sure, they might use an unfavorable move anyway to throw you off but once the initial surprise is gone you already know what to deal with.
As long as players always plan to adapt, no one should end up losing to the "random" factor too often, if at all.