First you need to understand will your form work with Javascript disabled or not. I recommend to disable Javascript in your browser and submit your form after that. If the form will work then all you need is just submit your data with WWW::Mechanize:

$mech->submit_form(
          with_fields => {
              login => "somelogin",
              password => "somepassword",
         },
);
[download]

If you can't submit your data from the browser with Javascript disabled you can either emulate same POST request in your code or you can use browser automation (like WWW::Mechanize::PhantomJS)