Didn’t you have trouble getting Skype Chat to work as the same time as a game? That’s usually poison.
There are programs that can record Skype just dandy, but I’ll put money they don’t work along along side a game.
Scroll down in this post.
https://forum.audacityteam.org/t/help-recording-with-a-mic-laptop-and-headphones/35068/1
Koz