You don't need TV for the capture part at all. What you need, is way to split or run-through video signal from 1st laptop. So the setup would be same as all of the other setups with PC with thing to be streamed and another machine for doing encoding.
Laptop#1 -> audio/video signal -> capture card or splitter -> TV // Laptop#2.
Edit: The capture card you have linked already has ability to be used as pass-through. So no need for splitter.
