[BUG] Cost not calculated for LLM Gateway interactions

Environment

• Platform (select one):
• [ ] Anthropic API
• [ ] AWS Bedrock
• [ ] Google Vertex AI
• [x] Other: LiteLLM Proxy 1.71.1
• Claude CLI version: 1.0.17 (Claude Code) <!-- output of claude --version -->
• Operating System: MacOS - Apple Silicon<!-- e.g. macOS 14.3, Windows 11, Ubuntu 22.04 -->
• Terminal: Warp.dev<!-- e.g. iTerm2, Terminal App -->

Bug Description

When using Claude Code with LiteLLM as the backend, token count, and costs, are not tracked.

Steps to Reproduce

1. Set up LiteLLM Proxy
2. Point Claude Code at LiteLLM Proxy (ANTHROPIC_BASE_URL)
3. Prompt Claude Code
4. Exit Claude Code <- Notice total tokens and pricing is 0

Expected Behavior

<!-- What you expected to happen -->
Spent tokens should be tracked and pricing info be updated.

Actual Behavior

<!-- What actually happened -->

Tokens:                0 input, 0 output, 0 cache read, 0 cache write

Additional Context

<!-- Add any other context about the problem here, such as screenshots, logs, etc. -->
Curl command to LiteLLM endpoint to verify token usage is included in the response:

curl "http://localhost:4000/v1/messages?beta=true" \
     --header "x-api-key: someKey" \
     --header 'anthropic-beta: claude-code-20250219,interleaved-thinking-2025-05-14' \
     --header 'anthropic-dangerous-direct-browser-access: true' \
     --header 'anthropic-version 2023-06-01' \
     --header "content-type: application/json" \
     --data \
'{
    "model": "claude-sonnet-4-20250514",
    "max_tokens": 100,
    "messages": [
        {"role": "user", "content": "Hello, world"}
    ]
}'
{"id":"msg_bdrk_014URjyq2jZ7Mrjfm1CpAtnp","type":"message","role":"assistant","model":"claude-sonnet-4-20250514","content":[{"type":"text","text":"Hello! Nice to meet you. How are you doing today? Is there anything I can help you with?"}],"stop_reason":"end_turn","stop_sequence":null,"usage":{"input_tokens":10,"cache_creation_input_tokens":0,"cache_read_input_tokens":0,"output_tokens":25}}