In the Gamefreak building on Akala Island in Heahea City (where you would get the Oval and Shiny charms), theres a guy who asks you which color you prefer between the 4 colors of the first-gen games. In US/UM only, if you have a pokemon that has been transported from a Gen 2 game in your party (not sure if it works with a pokemon from a gen 1 game, also it doesn't need to be Scyther specifically it can be anything, that's just what I sent up personally), the guy will make a comment and give you a full set of Apricorn balls.
Now I'm not sure if you have to have done anything before you can get the balls (obviously you need to have gotten far enough in Akala Islands trials to access the part of the city that Gamefreak is in), and again I'm not sure if the same pokemon can be used multiple times on different saves (also I might not even be thinking of the right guy..., but if you wanna try it out I don't have any personal attachment to this Scyther. Also I'm ready whenever you are. Pretty sure I still have you registered as a friend if you wanna do it tonight.