It is possible to record the switch screen with some camera. Obviously not the best quality
If you can’t afford a capture card then a camera is good to use, not good quality but it’s better than nothing.
You could also use a smartphone / tablet with a stopwatch to time the run.
Just download the app Frameperfect off the GooglePlay store, its what I use for splits and timing when not on a pc, it works great. You make your own splits and can name and put whatever pics you want for them. It also has most games already in it so just search Odyssey and it should already be on there. Then simply add splits and go for it!
Frame Perfect is great, I prefer it over Llainfair. I’ve been using it for a while, and spending the $2 to unlock extra features is worth it