I got this working! My problem was that the default server profile is “production,” which specifically disallows access to the Lucee admin pages when accessing from anything other than localhost. I was already planning on putting my Lucee service behind a reverse proxy, but I wanted to be certain it was working. The easiest way to was to set the environment variable BOX_SERVER_PROFILE=development in my docker-compose.yml and it worked beautifully.
Immediately after that, I ran into a bug where I was forced back to the login page on every click in the admin menu, and that was fixed by finally setting up the reverse proxy with proper SSL config.