refactor(search): /api/search/ask dispatcher route via llm-router

PR-2 of DS AI routing policy (2026-05-23, see plan
~/.claude/plans/document-server-ai-cheeky-reddy.md +
memory project_document_server_ai_routing_policy).

DS 의 모든 backend 호출이 llm-router :8890 단일 경유. 정칙 정합:
- 신규 RouterBackend (services/llm/backends.py) — alias 별 router POST
  + requires_gate 분기 (mac-mini-default 만 llm_gate FOREGROUND 보호).
- 기존 GemmaMacMiniBackend + QwenMacBookBackend = legacy 보존
  (DS_BACKENDS_VIA_ROUTER=false rollback safety only). 1주 후 별
  cleanup PR (PR-DS-Backends-Legacy-Cleanup-1) 로 폐기.
- get_backend factory dual-path (env flag) — backward-compat
  (gemma-macmini alias → mac-mini-default 매핑).
- search.py:457 Query pattern 확장: mac-mini-default|claude-cloud|auto
  추가. /ask/react 의 isinstance(QwenMacBookBackend) → hasattr
  duck-typing (RouterBackend + Legacy 모두 generate_with_tools 구현).
- SearchAskBackendConfig 에 router_url 신규 (env LLM_ROUTER_URL 또는
  hardcoded MVP default http://100.76.254.116:8890).
- docker-compose.yml fastapi env 에 LLM_ROUTER_URL +
  DS_BACKENDS_VIA_ROUTER 추가.

AIClient (_call_chat, call_triage, call_primary, call_fallback) 경유
path 는 별 PR (PR-AIClient-Router-Migration-1) — MVP scope C 채택,
회귀 risk 최소화.

Closure (즉시 fixture/matrix):
- factory smoke 6 alias (None/mac-mini-default/gemma-macmini/
  qwen-macbook/claude-cloud/auto) + 1 invalid (nonsense → ValueError).
- live 3 case: mac-mini-default 200 \"pong! 🏓\" + qwen-macbook cold
  502 upstream_502_primary=ConnectError + claude-cloud 503
  provider_not_configured.
- silent fallback 0 + direct M5/Mac mini socket 0
  (RouterBackend 만 router 호출).

Backup: ~/.local/share/ds-routing-pr2-backups/20260523/
(backends.py + config.py + search.py + docker-compose.yml).

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

app/core/config.py

@@ -36,18 +36,27 @@ class DeepSummaryBacklogConfig(BaseModel):
 
 
 class SearchAskBackendConfig(BaseModel):
-    """PR-MacBook-RAG-Backend-1: /api/search/ask backend dispatcher.
+    """PR-2 of DS AI routing policy ([[document-server-ai-routing-policy]], 2026-05-23):
+    /api/search/ask backend dispatcher 가 llm-router :8890 단일 경유.
 
-    backend 미지정 = Gemma Mac mini (settings.ai.primary 경로 그대로).
-    backend="qwen-macbook" 명시 opt-in = MacBook M5 Max mlx-vlm.server.
-    MacBook unavailable 시 503 + error_reason=macbook_unavailable (자동 fallback 없음).
+    - backend 미지정 / "gemma-macmini" / "mac-mini-default" → router 가 tier_b
+    - backend "qwen-macbook" → router 가 named upstream (M5 Max)
+    - backend "claude-cloud" → router 가 503 명시 (scaffold)
+    - backend "auto" → router 의 rule + LLM triage
+
+    Unavailable → BackendUnavailable → 503 명시 (silent fallback 0).
+    Rollback: DS_BACKENDS_VIA_ROUTER=false 로 legacy 직접 호출 path.
+    legacy macmini_url / macbook_url / macbook_model 은 fallback 시만 사용.
     """
 
+    # PR-2 신규: llm-router URL. 비면 env LLM_ROUTER_URL 또는 hardcoded default.
+    router_url: str = ""
+    # Legacy fields (DS_BACKENDS_VIA_ROUTER=false 시만 사용)
     macmini_url: str = "http://100.76.254.116:8801"
     macbook_url: str = "http://100.118.112.84:8810"
     macbook_model: str = "mlx-community/Qwen3.6-27B-8bit"
-    timeout_connect_s: int = 1
-    timeout_read_s: int = 30
+    timeout_connect_s: int = 5
+    timeout_read_s: int = 60
 
 
 class SearchAskReactConfig(BaseModel):