I believe the stuff that LWP does deals with HTTP headers, and in your case it looks like the info is in form fields. You're not likely to find a pre-made solution that fits this method. You could possibly roll your own with Authen::NTLM. The first step would probably be to log into the site using your browser and watch how it works using something like Paros or Charles.
in reply to LWP::UserAgent vs. NTLM Challenge/Response
Speaking of those tools, an alternative might be to watch if they set a cookie, like vcTheGuru suggested. If so, then you might be able to hardcode a spoofed cookie to trick the site into thinking you have one long never-ending session.
Even if neither of my ideas work out, I'd suggest trying one of the tools I linked to above. They'll help take the guesswork out of what you need to do.