LWP::UserAgent doesn't appear to set a cookie without a FQDN. Unfortunately, perlmonks.org isn't fully qualified. Use www.perlmonks.org or perlmonks.org. with the trailing dot. (If you look at HTTP::Cookies::add_cookie_header(), you'll see that it expects there to be two dots in the domain name.)
I had gotten as far as realizing that the cookie wasn't being set, and I had started looking at HTTP::Cookies before I left work, but I didn't catch this. That fixed it perfectly, thanks!
"We're experiencing some Godzilla-related turbulence..."