You can't. Pokemon Showdown! is it's own thing. Think of it as a separate battle simulator from the actual game. The closest thing you can get to it is that a program called pkHeX has a feature that lets you just put a Pokemon from showdown in to create, but you would need to mod your 3ds to have Homebrew, and that would be a whole other thing.
https://www.youtube.com/watch?v=VuyT6w1KinA <-- tutorial on how to get homebrew and use pkHeX. I can tell you that it works on 11.3.0-36 also, because I'm running it on that.
note: Always back up your save file!
(Also on the websites that ask for the 3ds version, put in the one you see on your system settings)
I hope I helped!