WWW::Mechanize does not have any DOM or Javascript facilities, so you have no other choice but to look at the Javascript yourself and emulate what the Javascript does in Perl.

If you're adventurous, you could look at what the CPAN has to offer on the topic, and use one of these solutions as basis. But you will still have to implement the browser DOM, which is quite some task to do properly.

perl -MHTTP::Daemon -MHTTP::Response -MLWP::Simple -e ' ;    # The  
$d = new HTTP::Daemon and fork and getprint $d->url and exit;#spider
($c = $d->accept())->get_request(); $c->send_response( new   #in the
HTTP::Response(200,$_,$_,qq(Just another Perl hacker\n))); ' #  web
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