docs: Add MLflow AI Gateway to LLM gateway configuration page

Request

Add MLflow AI Gateway as a supported LLM gateway on the LLM gateway configuration page, alongside the existing LiteLLM section.

What is MLflow AI Gateway?

MLflow AI Gateway (MLflow ≥ 3.0) is an open-source, database-backed LLM proxy built into the MLflow tracking server. It provides:

• Unified API across 20+ providers (OpenAI, Anthropic, Gemini, Mistral, Bedrock, Ollama, etc.)
• Encrypted secrets management — provider API keys stored on the server
• Fallback/retry, traffic splitting, budget tracking
• Built-in usage tracing
• OpenAI-compatible and Anthropic-compatible passthrough endpoints

How it works with Claude Code

MLflow AI Gateway exposes an Anthropic-compatible passthrough at /gateway/anthropic/v1. Configuration is straightforward:

export ANTHROPIC_BASE_URL=http://localhost:5000/gateway/anthropic
export ANTHROPIC_API_KEY=unused  # provider keys are managed by the gateway

This follows the same pattern as the LiteLLM pass-through endpoint section on the current docs page.

Context

We're from the MLflow team and are adding gateway integration guides across major agent frameworks. We already have merged/open PRs for LlamaIndex, AutoGen, CrewAI, Haystack, and Google ADK. We also have an example PR for the Claude Agent SDK at https://github.com/anthropics/claude-agent-sdk-python/pull/852.

Since the code.claude.com docs don't appear to be in a public repo, we're filing this issue instead of a PR. Happy to provide any additional content or review a draft.