Yet another broken SSL server which cannot do TLS1.0 or better. Browsers work around this kind of error by retrying with a downgraded SSL version.
Recent versions of LWP (base of WWW::Mechanize) use IO::Socket::SSL/Net::SSLeay by default and you can simply enforce a specific SSL version like this:

$mech->ssl_opts( SSL_version => 'SSLv3');
[download]

I've tested it and it works with this broken server.