### Check for existing issues

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

### What happened?

`x-litellm-overhead-duration-ms` response header and `litellm_overhead_time_ms` spend log field are not populated for requests to `/v1/messages` (Anthropic format) and `/openai/v1/responses` (OpenAI Responses API). They are correctly populated for `/v1/chat/completions`.

**`/v1/chat/completions` (Bedrock backend) — works:** ``` < x-litellm-response-duration-ms: 80.379 < x-litellm-overhead-duration-ms: 80.379 ✅ ``` `litellm_overhead_time_ms` in spend log: `80.379` ✅

**`/v1/messages` (Anthropic format, Bedrock backend) — missing:** ``` < x-litellm-response-duration-ms: ... # x-litellm-overhead-duration-ms: not present ❌ ``` `litellm_overhead_time_ms` in spend log: `null` ❌

**`/openai/v1/responses` (Azure OpenAI backend) — missing:** ``` < x-litellm-response-duration-ms: 14.966 < x-litellm-callback-duration-ms: 0.0 # x-litellm-overhead-duration-ms: not present ❌ ``` `litellm_overhead_time_ms` in spend log: `null` ❌

### Steps to Reproduce

```bash # Works — overhead header present curl -v -H "Authorization: Bearer <key>" \ -H "Content-Type…