Hi Ben, welcome to the forum!
With the current Live API it’s not possible to integrate with Live on that level, so unfortunately we can’t add an integrated recording feature like that.
However, it is possible to record Videosync’s output in another application, like Syphon Recorder. You can read more about this here.
Rob has has done some experimenting recording in an automated way using OBS, you can read more in this topic.
Let me know if you have any further questions!