What I would do, first, is to crank up a nice debugger (like Firebug for Firefox ...) in the web-browser, so that you can w-a-t-c-h the exchanges that take place for a successful login, or for an unsuccessful one, and also for logoff.   For example, are “authorization headers” used by this web-site, or does something else happen?   Does it (as expected) set a cookie?   Once you can clearly see what the target web-site expects and does, you can create LWP code to do that.   But first you have to understand the target.   Every one is typically a little bit, uhh, atypical.   ;-)