Hi Kazooie
LiveServerTestCase is a bit more lightweight than tools for using specific drivers. It doesn't tie into Selenium, or Playwright, or any other particular web browser library. Therefore I don't think there's going to be any way for LiveServerTestCase to know what the current page is, in order to save its contents.
I think, at this time, it would be interesting to see a third party package that implements what you're suggesting, for a particular web browser library. This way you can be more specific about what you support, and how.
Thanks,
Adam