Add weather caching, useless-tip redirect, clock widget, and IAM Bedrock auth

- Cache the latest fetched tomorrow weather and expose it via /api/system/weather/latest
- Add a header Clock widget showing date/time plus the cached weather summary
- Redirect useless-talk mentions from the weather bot to the useless-tips webhook
- Switch Bedrock auth from bearer token to standard IAM access key/secret

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

backend/app/core/weather_cache.py

@@ -0,0 +1,19 @@
+"""直近で取得した「明日の天気」データをメモリ上にキャッシュする。"""
+from datetime import datetime
+
+_latest: dict | None = None
+
+
+def set_latest(weather: str, pop: int | None, judgement: str, reason: str) -> None:
+    global _latest
+    _latest = {
+        "weather": weather,
+        "pop": pop,
+        "judgement": judgement,
+        "reason": reason,
+        "fetched_at": datetime.now().isoformat(),
+    }
+
+
+def get_latest() -> dict | None:
+    return _latest