Some accumulated wisdom and another question:
Some sniffing and blundering around on the net led me to the Crypt::SSLeay docs page, which says:

At the time of this writing, libwww v5.6 seems to proxy https requests
+ fine with an Apache mod_proxy server. It sends a line like:

  GET https://www.nodeworks.com HTTP/1.1

to the proxy server, which is not the CONNECT request that some proxie
+s would expect, so this may not work with other proxy servers than mo
+d_proxy. The CONNECT method is used by Crypt::SSLeay's internal proxy
+ support.
[download]

Sure enough, using Crypt::SSLeay style proxy specification for HTTPS, I get the login page I have sought so long.

The new problem? it doesn't appear to get decrypted. When I print out $a->content, I get gibberish. That's going to make it difficult to Mechanize the login.

Update:
It turns out that the silly headers I was passing to make my script look more like a browser were the problem. I made my script not pass them in the HTTPS request, and suddenly I had a normal web page.

Problem solved, thanks for your help, and I hope that somebody finds this useful down the road.