I need to be able to publish, from an automation, an MQTT message to both the internal broker and also to an external one. Best I can tell, I can only specify one broker in the configuration. Is there a way to do this in Home Assistant?
You can’t
What you can do is bridge your broker to the remote broker, and either re-publish every topic, or just certain ones.
Google for Mosquitto bridge and you’ll find many guides.