"Amusingly, that alone didn't do the trick for me on Windows because HTTP::Tiny is used as the download agent - and it's not capable of https transfers on a fresh build of perl from source."

Ah, sorry, I didn't appreciate that this was a perl you'd built yourself. Strawberry (I've no idea about AS these days) ships with the required IO::Socket::SSL and Net::SSLeay for direct SSL/https support within HTTP::Tiny.