Sadly, the only answer I can come up with will hog some extra bandwidth.
Firstly, use something like if (typeof FileReader !== "undefined"
or Modernizr to follow your normal flow for the browsers that DO support FileReader. Otherwise, POST the file via AJAX to some server-side script that echoes back the contents.
So for compliant browsers, you get to save yourself some bandwidth and for non-compliant browsers you have to take one for the team.