I dont think, what LWP can solve this problem, you can use IO::Socket module.

use IO::Socket;
$sock = IO::Socket::INET->new(PeerAddr => $proxy_server,
                              PeerPort => $proxy_port,
                              Proto    => 'tcp');
send($sock,"GET $url HTTP/1.0\n\n",'');
recv($sock,$answer,$length,'');
if($answer =~ m/.../)
{
...
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