I get the GET line. (Keep in mind that WWW::Mechanize is a subclass of LWP::UserAgent.)

use LWP::UserAgent qw( );
my $ua = LWP::UserAgent->new();
my $response = $ua->get('http://www.google.com/');
print($response->request()->as_string(), "\n");
[download]

GET http://www.google.ca/
User-Agent: libwww-perl/5.805
[download]

Update: hum, is that really what it sends? Is that even valid?
Update: yeah, it works if I send that raw. I'm not used to HTTP/0.9