progress of sorts: I have learned, using Wireshark, that the failure happens at the server, in the process of the hand-shaking between client and server involved in the establishment of an SSL connection. My script sends the usual hello message, and normally the other servers I connect to send back their own hello message. This one does not. The diaglog ends with my script sending it's first hello message to the server.
My challenge right now is two fold. 1) How to get wireshark to dump all the data it has collected (subject to my display filter) to a text file, so that I can send it to the tech guys at the company that owns the server. 2) How to set the level of SSL debugging in the SSL system used by LWP.