Streaming pass-through for native Anthropic /v1/messages and the /v1/responses
streaming iterator never set logging_obj.completion_start_time, so
_success_handler_helper_fn fell back to completion_start_time = end_time.
Downstream TTFT consumers (Prometheus, OTEL, Langfuse, Admin UI, spend logs
completionStartTime) then reported time-to-first-token equal to total request
duration.
Stamp completion_start_time on the first chunk in PassThroughStreamingHandler.
chunk_processor and BaseResponsesAPIStreamingIterator._process_chunk, mirroring
CustomStreamWrapper for /chat/completions.
Resolves LIT-4185
Co-authored-by: yucheng <yucheng@yuchengs-MBP.localdomain>
