♻️(backend) extract LiveKit connection info generation function

Extract serialization logic for LiveKit server connection data to make it
reusable across endpoints. Function naming will be improved in future
refactoring when utility functions are moved to a proper service.

src/backend/core/utils.py

@@ -81,3 +81,21 @@ def generate_token(room: str, user, username: Optional[str] = None) -> str:
     )
 
     return token.to_jwt()
+
+
+def generate_livekit_config(room_id: str, user, username: str) -> dict:
+    """Generate LiveKit configuration for room access.
+
+    Args:
+        room_id: Room identifier
+        user: User instance requesting access
+        username: Display name in room
+
+    Returns:
+        dict: LiveKit configuration with URL, room and access token
+    """
+    return {
+        "url": settings.LIVEKIT_CONFIGURATION["url"],
+        "room": room_id,
+        "token": generate_token(room=room_id, user=user, username=username),
+    }