There is a Perl module to provide access to this thing.Have your tried this?
http://search.cpan.org/~exobuzz/MediaWiki-API/

Update oh I see you have that one. I was also trying with LWP and tried enabling cookies and a few other things. I can login to other secure sites, but something is weird about this one. I don't know why.

Update:Still haven't figure this out, but there appears to be a couple of flavors of SSL: SSL2, SSL3. SSLeay-0.81 has SSL3. Crypt-SSLeay 0.061 requires SSLeay 0.81. I have Crypt:SSLeay 0.57. I am clueless as to how to get these later versions for Perl or if they are even necessary. But I thought I'd mention this possibility as it may help others "in the hunt".