Does your CRT accept RGB SCART?
Assuming your CRT takes RGB SCART or you can easily secure a cheap one that does, I'd get a RGB SCART cable for the Wii and send that to a matrix switchbox like the gscartsw lite or Shinybow 5525. Send one of the outputs from the switchbox to the CRT. Then send the other output to a cheapo SCART to HDMI upscaler and finally to a capture device of your choice. The switchbox will be the big ticket item here but you will be able to add many other RGB SCART compatible systems to your setup and easily stream those as well.
If the switchbox is more than you want to spend or RGB SCART is not possible, I think you are looking at connecting your Wii to a component splitter. Send one output to your CRT and the other to a used El Gato Game Capture HD that still has its component video adapter dongle.