[backup-service] Propagate 'BlobClient' to handlers

Summary:
Part of https://linear.app/comm/issue/ENG-2688/backup-service-reuse-single-blob-client-instance

Citing https://phab.comm.dev/D6167?id=20696#inline-41664 and Tonic docs - the Channel is encouraged to be cloned and so the BlobClient instance is.
I propagated it to all places where it would be needed, the same way as DynamoDB DatabaseClient. It's not yet used, though.

Depends on D6384

Test Plan: Backup service compiles.

Reviewers: varun, jon, michal, tomek

Reviewed By: tomek

Subscribers: ashoat, atul

Differential Revision: https://phab.comm.dev/D6385