Get a capture card or point a camera at your Switch. You'll need to run the timer on a PC, yeah.

Hard to say, everyone has their preferred way to capture HDMI. Elgato, Avermedia, even Razer all have HDMI capture cards and there’s other brands too. Each person will have their preference, though an internal is the best way to go because internal PCI cards won’t have as much delay as an external over USB.

Look around, get some opinions, read some articles, etc. and find out which card seems like the best for you. Everyone has their own opinions on which they prefer.

Which game you interested in running?

People tend to shit on Elgato but I've had mine since I started streaming (coming up on 3 years) and I've never had the slightest problem with it.

I hope at some point Nintendo decides to increase the video capture time. I would love to be able to record runs on the go. But, unfortunately so far all we can do is dock it with a capture card.