Using Windows 2000, ActiveState perl v5.6.1 and WWW::Mechanize (version 0.43)-- I cut and pasted your code and it runs perfectly on my system. So, perhaps the problem is in your version of Mechanize, or perhaps you're missing something else on your system. When I run the code through my debugger, it does tap into LWP::UserAgent. Could you have a problem there? (Seems unlikely, but leave no stone unturned.)
I know this isn't much help. But don't give up hope, the code can work.