Some proxies also require that you send it a username/password in order to let requests through. You should be able to add the required header, with something like this:

 use LWP::UserAgent;

 $ua = LWP::UserAgent->new;
 $ua->proxy(['http', 'ftp'] => 'http://username:password@proxy.myorg.com');

 $req = HTTP::Request->new('GET',"http://www.perl.com");

 $res = $ua->request($req);
 print $res->decoded_content if $res->is_success;
Replace proxy.myorg.com, username and password with something suitable for your site.