### Check for existing issues

- [x] I have searched the existing issues and checked that my issue is not a duplicate.

### What happened?

Summary

/chat/completions streaming correctly records completion_start_time (time of the first chunk), so TTFT in the Admin UI / Prometheus / OTEL is accurate.

/v1/messages and /v1/responses do not, in two different ways:

/v1/messages — the fix from #25599 (merged via #25828) only patches the plain streaming path (BaseAnthropicMessagesStreamingIterator.async_sse_wrapper). If any registered callback overrides async_should_run_agentic_loop (e.g. the built-in websearch_interception / compression_interception callbacks, or any custom CustomLogger implementing that hook), the stream is wrapped in AgenticAnthropicStreamingIterator instead, which never sets completion_start_time at all. /v1/responses — completion_start_time is never set anywhere in the Responses API streaming path. This triggers the end_time fallback on every streaming request, unconditionally — no special config required.

In both cases, litellm_logging.py's fallback kicks in:

python# litellm/litellm_core_utils/litellm_logging.py if self.completion_start_time is None: self.…