@Mitchell Wortman Thanks for reaching out to Microsoft Q&A, apologize for any inconvenience caused on this.
Based on the information shared, My understanding of your ask is to copy the search index of one cognitive search instance to another to implement the Business continuity and diaster recovery for your search service.
As mentioned in the documentaiton, Cognitive Search doesn't provide instant failover in the event of an outage.
If you want to make your search service available across geographic regions, you can choose specific scenarios from here and implement your solution based on your requirements.
Refer to this documentation, for more information about synchronize the data across multiple services.
Feel free to reach back to me if you have any further questions on this.