How to record a video with two different audio tracks?

Question

Video broadcast

You need to record a video with two audio tracks or two videos with different tracks, and broadcast one of them over the network during recording. At the same time, people who are far from technology should be able to start-stop everything looking at the instructions. Now the video signal comes from an analog camera, the sound from two microphones through a mixer. One microphone is used all the time, sometimes translations or comments are spoken in parallel in the other.
Now the recording goes like this: there are two tuners in the computer, and a video signal splitter is connected to the camera. In the case of one track, MS Expression Encoder is used, which takes the video from one tuner, and the sound from the computer. If it is necessary to record the second track, the recording from the second tuner is switched on in parallel with its bundled software, the second microphone is also connected to the second tuner.
First of all, I want to get rid of leaving only one tuner, and also replace Expression Encoder with something that supports modern codecs (perhaps an old version is used) and does not require sole access to the tuner. I ask you to throw in software for recording and broadcasting, now only VLC comes to mind.