It's true that VSCode Developer Tool cannot inspect extension network because it's running under separate process called the Extension Host. You can take a look here but it seems complicated to setup https://stackoverflow.com/a/67099176/3789481
A simpler approach is to use Fiddler Classic as a forward proxy. By doing this, all API requests from VSCode will go through Fiddler, allowing you to inspect them, including HTTPS requests through TLS Inspection.
Fiddler setup:
After that, you can inspect for any *.githubcopilot.com directly, e.g. inline suggestion
VSCode will send prompt and get response back as Content-Type: text/event-stream
You can also inspect GHCP Chat with that same endpoint