Send exceptions that happen in Handler#process to Sentry

What does this MR do and why?

We recently had failures that weren't sent to Sentry. This made discovering the problem harder than expected: !2569 (comment 1702653256)

Expected impact & dry-runs

These are strongly recommended to assist reviewers and reduce the time to merge your change.

See https://gitlab.com/gitlab-org/quality/triage-ops/-/tree/master/doc/scheduled#testing-policies-with-a-dry-run on how to perform dry-runs for new policies.

See https://gitlab.com/gitlab-org/quality/triage-ops/-/blob/master/doc/reactive/best_practices.md#use-the-sandbox-to-test-new-processors on how to make sure a new processor can be tested.

Action items

• If adding environment variables for reactive processors, update config/triage-web.yaml and .gitlab/ci/triage-web.yml
• (If applicable) Add documentation to the handbook pages for Triage Operations =>
• (If applicable) Identify the affected groups and how to communicate to them:
  • /cc @person_or_group =>
  • Relevant Slack channels =>
  • Engineering week-in-review