The best way to record it quality wise is buying a 3DS capture card (which are pretty expensive). However, if you just need to record gameplay you can simply just point a camera to your 3DS while playing.
Yeah, on top of being expensive af 3DS capture cards are basically indefinitely out of stock so...
But we accept any video quality as long as it's good enough to let us see what's going on in the game, so just pointing a camera at your top screen (bottom screen isn't even needed) is enough. And just a reminder that only really good times are required to provide video evidence (anything under 1:23:00 in All Cups, and the time limit for other categories can be found in the rules of these categories), so you may not even need any of this to have your runs verified ^^
Some ideas if your phone has a camera: Resting on top/off the side of a table. Set it on your chest, use t-shirt to keep stable (what I do). Have someone else record. Put a small box/misc object in your lap, lean the phone against that.
