I put in a lot of time to come up with a solution on a previous occasion.

Update: Here is a module that adds a post function to Win32::IE::Mechanize.

=pod

   Win32/IE/Mechanize/Post.pm

   Adds function post to Win32::IE::Mechanize
   if it doesn't already exist.

   usage:
      use Win32::IE::Mechanize::Post;
         -or-
      use Win32::IE::Mechanize::Post ();

      The post function accepts for arguments anything
      HTTP::Request::Common::POST would accept.

      $ie->post('http://www.example.com/survey.cgi', [
         name   => 'John Doe',
         email  => 'doe@example.org',
         gender => 'M',
         born   => '1976',
         perc   => '3%',
      ]);

      # multipart/form-data
      $ie->post('http://www.example.com/survey.cgi',
         Content_Type => 'form-data',
         Content => [
            name   => 'John Doe',
            email  => 'doe@example.org',
            gender => 'M',
            born   => '1976',
            init   => ["$ENV{HOME}/.profile"],
         ],
      );

=cut

use strict;
use warnings;

use Win32::IE::Mechanize qw( );

package Win32::IE::Mechanize;

use HTTP::Request::Common  qw( );
use Win32::OLE::Variant ;# qw( Variant VT_* );

if (!*post{CODE}) {
   *post = sub {
      my $self = shift;
      my $agent = $self->{agent};

      local $HTTP::Request::Common::DYNAMIC_FILE_UPLOAD = 0;
      my $request = HTTP::Request::Common::POST( @_ );

      my $uri = $request->uri;
      ($uri = $self->uri 
         ? URI->new_abs( $uri, $self->uri->as_string )
         : URI->new( $uri )
      );

      my $headers = $self->_extra_headers( $uri )
                  . $request->headers_as_string( "\015\012" );
         
      $agent->navigate({
         URL      => $uri->as_string,
         Headers  => $headers,
         PostData => Variant( VT_UI1, $request->content ),
      });

      $self->_wait_while_busy;
   };
}

1;
[download]

Untested. Let me know if there are any problems.

Next step: testing it and submitting it as a patch.

The only, ugly, solution I know would be to replace the HTML in the current page with a <form action="$target_url" method="POST">...</form>, but hopefully there is a better way...