It's my understanding that LWP is good for grabbing a single web page (or file), but not for automating interaction with web pages like what you're wanting to do.

I believe that you'll probably want to use WWW::Mechanize. That module is used "for stateful programmatic web browsing, used for automating interaction with websites", which sounds like what you're trying to do. If you're needing JavaScript support, I believe that WWW:Mechanize::Firefox would enable support for JavaScripts.