I tend not to use HTTP::Request directly unless I have to. I use LWP::UserAgent's post method thusly (untested):

my $ua = LWP::UserAgent->new(keep_alive => 1, requests_redirectable =>
+ [qw/GET POST HEAD/]);

my %args = (
    foo => "bar",
);

my $response = $ua->post($form_url, \%args);
[download]

You might also find that the POST request gets redirected, in which case the requests_redirectable should help.
If this doesn't fix it, we're going to need more information..