I tried your code, too. It actually looks like that :

my $ua = LWP::UserAgent->new(
            'agent' => basename($0),
            'timeout' => $TIMEOUT,
            );

my $cookie_jar = HTTP::Cookies->new(
    file            => "perlcookie.txt",
    autosave        => 1,
    ignore_discard    => 1,
    );
$ua->cookie_jar( $cookie_jar );

my $client = REST::Client->new( {
        host    => $url,
        timeout    => 10,
        useragent    => $ua,
        follow        => 1,
    } );

$client->addHeader ("content-type", 'application/x-www-form-urlencoded
+');

$res = $client->POST(
            "j_security_check",
            "j_username=$username j_password=$password",
        );
print Dumper ($res); 
$res = $client->GET("api/14/system/info");
$rc = $client->responseCode ();
print Dumper ($res);
exit ( $rc);
[download]

And cookie file contains that

#LWP-Cookies-1.0
Set-Cookie3: JSESSIONID=node0npoji3aifa9r1c2xuxr6xvppe221132.node0; pa
+th="/"; domain=[SNIP]; path_spec; discard; HttpOnly; version=0
[download]

Unlike previous attempts, all HTTP transactions look good with this version of the code

SOLVED (almost) !

By default, LWP::UserAgent forwards only GET and HEAD requests. Adding POST gives me the login page.

See requests_redirectable.