I thought of another solution but it is a little complex. Set up some fake HEAD requests with a max-age: 1 response header to coerce the browser into re-fetching them. You should then receive an if-modified-since header from any modern browser like so:
If-Modified-Since: Sat, 29 Oct 1994 19:43:31 GMT
Just be careful not to send any Last-Modified headers with the first response because
To get best results when sending an If-
Modified-Since header field for cache validation, clients are
advised to use the exact date string received in a previous Last-
Modified header field whenever possible.
Note the "whenever possible" disclaimer. This, and other parts of the header description imply that the client will use its own clock when it doesn't know anything of the servers.
With the right combination of headers this could actually work very well.
EDIT: Tried some tests with FF but couldn't find a valid combination of headers to trigger an if-modified-since in client time. FF only sends the header if it got a last-modified header previously and then it just reflects the value back (even if it isn't a valid date).