Instead of your version:

  $agent->{req} = $request;
  $agent->_do_request;
[download]

You might want to try:

  $agent->request($request);
[download]

This seems to be the new internal method instead of _do_request. Apparently it takes in the $request object as a parameter, rather than requiring you to set it manually with $agent->{req}.

Note: this is completely untested. I just looked at WWW::Mechanize's perldocs and made an educated guess. :)