Again: enable strict and warnings, especially when you're looking for a bug!

The code should work now. But let's do some debugging:

#$URL and $xml_req initialised
print "Content-type: text/plain\r\n\r\n";
  #rather use CGI.pm anyway

print "URL: $URL\nXML: $xml_req\n-----\n";

my $ua = LWP::UserAgent->new();

print "Useragent: $ua\n";
  #will print reference representation

my $req = POST $URL,['form_param'=>$xml_req];

print "Request:\n",$req->as_string(),"\n-----\n";

my $result = $ua->request($req);

# rest is clear...
[download]

The problem is the generated Request (speculation). You can be sure after reading the documentation of HTTP::Request::Common

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