It seems like you have managed to pose a question where few people have experience.
If you were to ask for audio recording, you would get a lot of responses. Any DAW would do that. Audacity is free, not really a DAW, but fine for recording audio if you dont do fancy stuff. I like Reaper, which is an affordable DAW with a lot of capabilitis and a long free trial.
And to edit together the two videos, any "multitrack" video editor would work. ShotCut is free, but not super stable. The Reaper DAW can actually edit video, but is not super easy to use - it mostly makes sense if you record your audio there and add video as a side thing. I believe that Power Director is a popular video with multiple tracks. I dont know if that or any other editors allows for recording in the app. Otherwise, with your gear you would want an app on your PC that records video and allows for external mic. I read online that the Windows 10 camera app has some issues on that point, but you can test it.
You want to record video directly into and editor while watching playback of the first track, right?
I never heard of any PC software for that. But I think that is what some people do with a iPhone app called Acapella or something like that. I dont know if it comes for PC.