### Check for existing issues

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

### What happened?

Streaming `POST /v1/messages` (Anthropic-format endpoint) routed to an `openai/`-provider backend returns HTTP 200 successfully and streams a correct Anthropic-shape SSE body to the client — but **no logging callback (`StandardLoggingPayload`) is ever emitted** for the request. The request is billed by the provider but recorded with 0 tokens / $0 cost, so it's invisible to any success-logging integration (spend tracking, S3/Datadog callbacks, etc). No exception is raised anywhere — this is a silent gap, not a crash.

This affects **both** of the custom Anthropic-messages streaming adapters:

| Adapter | Backend models | Streaming wrapper | |---|---|---| | `litellm/llms/anthropic/experimental_pass_through/adapters/` | chat-completions-API models (e.g. `gpt-4o-mini`) | `AnthropicStreamWrapper` | | `litellm/llms/anthropic/experimental_pass_through/responses_adapters/` | Responses-API models | `AnthropicResponsesStreamWrapper` |

### Root cause

Both `AnthropicStreamWrapper` and `AnthropicResponsesStreamWrapper` drain the inner provider SSE stream …