Have you tried using the website with Javascript disabled in your browser? Have you looked at the difference in headers that your browser sends as opposed to what WWW::Mechanize sends? Have you looked at the web page source that is used to determine whether you have Javascript enabled or not?

Also, see the documentation of WWW::Mechanize, which tells you that there is no hidden magic by which you can "enable" Javascript in WWW::Mechanize.

There are Javascript-enabled alternatives to WWW::Mechanize, for example WWW::Scripter and WWW::Mechanize::Firefox.