Main options for recording are to point a camera at the screen, or to buy an HDMI capture card.
If you want a timer on the recording you'll need to use third party recording software such as OBS, but you don't need to have that - you can always retime afterwards, and the mods check the timing anyway.
I use OBS, a cheap capture card and Livesplit One (since Window sharing bugs on Ubuntu and I can't crop)
I just hook up a cheap Mirabox capture card to OBS and stream off that
you can always just point a phone camera at your TV screen. then to retime you can upload it to youtube and use this website https://mk8dx-speedrun-timer.azurewebsites.net to retime your run as accurately as you feel you need. it's quite easy to go frame by frame with it and you can just look at the video to know when to mark the beginning and the end. cheers and good luck!
I use a Genki Shadowcast and a HDMI splitter. The Shadowcast Capture card has a little latency so it goes to my laptop to record while the splitter sends a copy of the Switch's output to my TV. Then I use OBS to record me playing with a timer. Hope this helps!
But what if the audio is bugged in your run?
I noticed something strange in Music Park