You haven't shown any (Perl) code so it's hard to help you with that. Let me recommend WWW::Mechanize, which emulates a browser far more faithfully than LWP::Simple (and LWP::UserAgent) does. What WWW::Mechanize doesn't do is interpret JavaScript, so you will still have to do that yourself.

The easy way to approach JavaScript is to use the Mozilla Live HTTP Headers extension for FireFox, trace what gets sent, and then replicate that with WWW::Mechanize, sending the GET or POST requests as needed. Especially in the JS snippet you posted it seems that they use some base64 conversion and some other crypting to make the data transfer "more secure", so your Perl code will have to make some effort to replicate that.