refactor(db): use shared pooled DatabaseClient singleton instead of per-call instances

- Replace get_db_client() creating new DatabaseClient on every call with a
  module-level singleton initialized once at startup via init_db_client()
- Add init_db_client() and close_db_client() lifecycle functions called
  from FastAPI lifespan handler
- Migrate all DatabaseClient methods from legacy self.connect()/self.conn
  to pooled self.get_conn() context manager for thread-safe connection reuse
- Pool is properly torn down on application shutdown

Closes leeworks-agents/SPARC#7

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

SPARC/auth.py

@@ -146,11 +146,36 @@ def decode_token(token: str) -> Optional[TokenPayload]:
         return None
 
 
+# Shared database client singleton, initialized at startup via init_db_client()
+_db_client: DatabaseClient | None = None
+
+
+def init_db_client() -> None:
+    """Initialize the shared database client. Call once at app startup."""
+    global _db_client
+    _db_client = DatabaseClient(config.database_url)
+    _db_client.connect()
+
+
+def close_db_client() -> None:
+    """Close the shared database client. Call at app shutdown."""
+    global _db_client
+    if _db_client:
+        _db_client.close()
+        _db_client = None
+
+
 def get_db_client() -> DatabaseClient:
-    """Get database client for auth operations."""
-    client = DatabaseClient(config.database_url)
-    client.connect()
-    return client
+    """Get the shared pooled database client for auth operations.
+
+    Returns the module-level singleton DatabaseClient. If not yet initialized
+    (e.g., during tests), creates a new instance as a fallback.
+    """
+    global _db_client
+    if _db_client is None:
+        _db_client = DatabaseClient(config.database_url)
+        _db_client.connect()
+    return _db_client
 
 
 async def get_current_user(