I'd say you should try Fate on for size, and ignore anyone who suggests a D&D-adjacent game. It's been a while since I've watched Sword Art Online, but what I do remember tells me that if you want your game to resemble what Sword Art Online actually is, in the watching, then you want something that can handle an anime plotline without all that much actual fighting in it, and no, Dungeons and Dragons and its imitators aren't what you want for that. Another game that might get brought up is Anima Prime, which is great for doing shonen anime with big, bombastic fight scenes, but I recommend against it here because SAO isn't a shonen anime, and you'd still end up working against the game to do what you want, which is either playing the game wrong, or playing the wrong game, and either way you're probably going to be feeling frustrated.