Let's take this one step at a time. You need a web client that handles HTTPS. Either LWP or HTTP::GHTTP will work. (And by the way, LWP *is* libwww.) Then you need to get at a page that requires a login. This is also no problem. Just make your program go to the login page, get the URL it returns to you, and follow it. Basically, you follow the same steps you would in a browser but you do it in your program.

You need to make an attempt to learn one of these HTTP client modules. If you get stuck, you can ask questions here. Just make sure they're specific, and not "write my program for me" kind of questions and you'll get plenty of help.