Log exceptions with the error tracking module

What does this merge request do and why?

This MR consolidates the logging for the exceptions with the new error tracking module.

Previously, there were some issues that:

• Correlation ID is not recorded in the error log.
• Multiple exceptions can't be attached to the API log.
• Backtrace is missing therefore we have no idea where the root cause is.

which resulted in an observability issue like #396 (comment 1710671986)+ and Improve observability of Duo Chat with v1/agent... (#371 - closed).

How to set up and validate locally

Numbered steps to set up and validate the change are strongly suggested.

Merge request checklist

• Tests added for new functionality. If not, please raise an issue to follow up.
• Documentation added/updated, if needed.