Hi Daniel, 
Using the Genesys API's is the easiest way to perform this action. We make use chat transcripts in RAW data extracts for our clients as well as using chat transcripts to automatically be sent, if requested, to clients after a chat is completed.  
Kind Regards

Yes, there are various ways to download in bulk through APIs.

Take a look at:
https://help.mypurecloud.com/articles/about-the-aws-s3-recording-bulk-actions-integration/

That is one option among other APIs as well.