Is there an API (or an alternative), to call the same extension overlay information that I would see in my browser?

For what it’s worth, I’m not planning on playing it back. I know that would be very difficult, but all I want to do is any generic way of storing the data that my browser receives on my computer in a file.

But if you aren’t sure of a way to do that, no worries, I know it might not be particularly supported.