[services] Backup - Blob Get Client - Refactor put write

Summary:
Depends on D5093

Linear task: https://linear.app/comm/issue/ENG-1734/improve-error-handling-system-in-the-blob-client

Using a new way of error handling in the write method of the put client.

Test Plan:

cd services/backup/blob_client
cargo check

after running localstack, blob and backup (in this particular order), this still works:

yarn run-performance-tests backup

Reviewers: jon, max, tomek, varun

Reviewed By: jon, tomek

Subscribers: ashoat, tomek, atul, abosh

Differential Revision: https://phab.comm.dev/D5094