I expected LWP::UserAgent to just be able to get the JSON document with its transparent cookie handling facility, but it can't. I've checked and it does seem to be sending the cookie, but again, I just get the login page.

:) There are other headers besides cookies, even multiple cookies, webservers freely discriminate on all data supplied

Is there a way to grab the cookie jar off the LWP::UserAgent object and hand it over to a WWW::Mechanize object?

you could use the cookie_jar accessor, it probably won't solve your problem above