Content scripts can access the window object of the page in which they're injected, so that shouldn't be an issue, I don't believe - if the page in which a content script is running can remove data from localStorage, the content script itself can, too.
One of the reasons I use that extension is that it deletes things a short while _after_ the last tab for a site is closed (so they undoing the close lets me keep going). In that case there's no content in which to run scripts.
