So a coworker suggested I try accessing my Plex instance by internal IP, not internal hostname. And that... worked. When using hostname, the way I always had for everything else, it'd show that I didn't have access to the server.
I had the same "", but after restarting all the web interface says is Connecting. Then the server just doesn't show up in the list. <sigh> Might just need to blow up my whole config and reimport everything.