The mode page talks about unlocking colors in Gallery, unless ArcSys is going hella balls on us and making you unlock anything that isn't player 1 and player 2, it seems like at the very least some of the extra palettes will be unlocked in-game. It's possible that 13-16 (reference, BB, Persona, Extra) could be DLC, but with the way that page is phrased, it seems unlikely that they all will be.
Unless they wanted to go hyper-balls on us and make you buy the colors and then go get them from the Gallery afterwards, but that'd just be