I would like to speedrun this game, but I have no recording setup and I cannot afford a capture card. Would split proof be enough or do I have to record? If I do have to record, does anyone know a setup for it?

A phone stand is a good place to start. If you don't have a stand, maybe try a Lego setup to prop it up? Might seem like a bit of hassle but you can get relatively good results. The other option would be NTR Streamer if your 3DS is on a firmware where you can have CFW (though if you're on the latest firmware, this isn't an option for you).