SingleFile [1] is supposed to be able to do that (more or less) by enabling these settings in the options page:

- User Interface > open saved page in a new tab

- User Interface > auto-close the tab after the page is saved

- Auto-save > auto-save on tab discard

Then you have to select in the context menu of the extension: Auto-save > Auto-save all tabs (or Auto-save unpinned tabs)

[1] https://github.com/gildas-lormeau/SingleFile