Refactor gateway logging flow to centralized middleware pattern

[wauputr4]

Context

PR #48 already moved request-log persistence off the response path and stopped hardcoding method/path in the logging helper.

The remaining work is a maintainability refactor: the gateway still has several manual completion-log call sites in crates/mizan-api/src/gateway.rs, which makes future endpoint additions harder to keep consistent.

Proposed changes

• Centralize gateway completion logging behind one helper or layer boundary so handler branches do not repeat completion plumbing.
• Keep the request-log schema and existing fields unchanged.
• Preserve current success/error semantics for chat completions.
• Make the logging path reusable for future non-chat gateway endpoints.

Acceptance Criteria

• Completion logging is centralized instead of repeated in each branch.
• Success and error paths still emit the same request-log fields.
• Existing async persistence behavior remains intact.
• Add tests or integration assertions for at least one success path and one error path.
• Document the tradeoffs if a middleware/layer pattern is introduced.

Notes

• Do not re-open the async persistence or method/path parameterization work already landed in PR feat: add request logging and admin audit foundations #48.
• This issue is a follow-up refactor for maintainability, not a blocker for the alpha release boundary.