You could get 3 Y splitters for RCA and split it right before your dazzler run one set to the TV and the other to the dazzler.
Sorry dazzler is in my phone's dictionary but the actual name is not
If you can get a capture card that works for you for under 50$, then thats all well and good. Most of the time with this type of thing, when you go cheap you end up with something that performs poorly or doesnt work at all.
Ive been streaming Super Mario Maker on my Wii U with the Elgato HD60 that I bought, and it works really well and looks great. No lag and broadcast in 60fps. I got that used for 90$, very much worth it.
I plan to buy the el gato game capture HD card to stream retro games. The game capture HD is only like 50-80 used. The adapter cable is like 20$ from the elgato website.
I can understand if you might want to stream while playing on your CRT monitor, but if you are serious about streaming I would just use a monitor with hdmi.