[DOCS] Document Claude Code pass-through mode, system_prompt_claude_code and extra_args

[edenreich]

Summary

Since inference-gateway/cli#749, Claude Code subscription mode (claude_code.enabled: true) is a true pass-through and the docs need to reflect the new behavior and configuration surface:

• infer no longer injects its system prompt, context blocks, or system reminders in Claude Code mode - claude runs with its own defaults and native tools. Infer prompts/reminders/tools config does not apply in this mode.
• claude's tool calls are no longer re-executed locally; claude's own streamed tool results are carried through.
• Native TaskCreate/TaskUpdate calls are mirrored as a synthesized TodoWrite view at the output layer only (headless stdout, consumed by infer-action); the stored conversation keeps the original calls.
• New prompt setting: prompts.agent.system_prompt_claude_code (empty by default; when set it is passed via --append-system-prompt, keeping claude's built-in prompt intact). Env var: INFER_PROMPTS_AGENT_SYSTEM_PROMPT_CLAUDE_CODE.
• New setting: claude_code.extra_args - extra arguments appended verbatim to the claude CLI invocation. Flag: --claude-code-extra-args; env var: INFER_CLAUDE_CODE_EXTRA_ARGS (comma/newline-separated, env wins over flag).

Example:

claude_code:
  enabled: true
  extra_args:
    - --max-turns
    - "5"

prompts:
  agent:
    system_prompt_claude_code: "Always answer in English."

Affected pages: Claude Code mode page and the CLI configuration reference (prompts + claude_code sections).

Source PR: inference-gateway/cli#749

Acceptance Criteria

• Claude Code mode page documents the pass-through behavior (no infer prompt/reminders/tool re-execution; TodoWrite mirroring is output-layer only)
• prompts.agent.system_prompt_claude_code documented with its env var and append semantics
• claude_code.extra_args documented with flag/env var and precedence (env > flag > config)