It was the bloody power source! I didn't bother hooking it up to a decent power brick because I figured the Pi zero didn't need as much juice as my Pi 3... Idiot... Thanks for your inputs guys.
@gtoal I don't see how this applies to the topic at hand - the poster specifically said it's a non-W(ifi) model and they don't try to use wireless at all.
I saw this thread, so I set up lr-hatari, got it working, and updated the docs, mostly just from the core's github readme. It worked pretty much first try, but the docs previously only covered Hatari standalone.
One thing I am not clear on is how you get the core to look at different BIOS files in order to emulate the STE or Falcon, but I haven't tried that yet.