Pre-restriction would have to disallow running javascript in that case. Otherwise an app could save data to localstorage, refresh the page programmatically (or wait for the user), and then transmit.
Or rather have the browser treat the no-network session period with private window logic where the session is deleted if the no-network browsing session ends or dies (e.g. incognito puts a virtual session data container in RAM). That would even allow the app data (prior to triggering the mode) to stay cached without risking side channels with future resource loads.
The whole premise was to return to the pre-web 2.0 days but have the option for a user to enable modern web app capabilities. Which your idea would be a step towards.