Correct Option is A.
Have the external system subscribe to a custom Platform Event that gets fired with EventBus.publish()

This allows:
Full control over event schema.
Custom data like error messages, job context, stack traces, etc.
Reliable communication via CometD or Webhook listener on the external side.

Only custom Platform Events can be published using EventBus.publish() to notify external systems.
Standard events can't be fired manually, and no automatic event exists for Apex exceptions.

Correct answer is C ,

No publishing is necessary. Have the external system subscribe to the standard BatchApexErrorEvent by implementing Database.RaisesPlatformEvents on the batch