If you're going to use raw LWP, you need to create a separate HTTP::Request object and set its ->referer property, then pass that to the LWP::UserAgent:

my $req = HTTP::Request->new();
$req->method('GET');
$req->url($uri);
$req->referer($referer) if $referer;
$content = $http->request($req);
[download]

Or you may want to take a look at WWW::Mechanize if you want an all-in-one package. Its docs include referer manipulation in one of the examples, so it would appear to be capable of this.