I solved this problem by requesting a TLS certificate from my company's Windows domain CA for my machine's hostnames on the internal network (machinename and machinename.company.com - both added to the same cert via Subject Alternative Name). Then I mapped those domains to 127.0.0.1 in /etc/hosts so it works even when I'm not on the internal network.

From there it's simple to host the front-end app I work on using a node.js script.