Thanks ikegami, but setting the domain param to -domain => domainname.com replicated the failed result. I tried forcing -domain => www.domainname.com and it wouldn't even write the cookie. So, basically, no domain param seems to work, which strikes me as strange.
And unfortunately, the redirect will not work because I'm hosting multiples sites all using the same code with relative paths and HTML::Template.
That said, I tried a redirect in the HTML, but even that didn't work unless the URI was specifically domainname.com/admin/index.html, which also seems strange.
Update: See my resolution in the above OP