✨(backend) extend live synchronization to lobby access level updates

Extend the existing live synchronization mechanism beyond room
configuration to also include lobby access level changes.

This ensures that all owners and admins sharing a room maintain a
consistent and up-to-date view of room state in the frontend,
including configuration and access control updates.

src/backend/core/api/viewsets.py

@@ -309,12 +309,20 @@ class RoomViewSet(
         """Persist the room update, then sync metadata to LiveKit."""
 
         old_configuration = serializer.instance.configuration
+        old_access_level = serializer.instance.access_level
+
         room = serializer.save()
 
-        if room.configuration == old_configuration:
+        if (
+            room.configuration == old_configuration
+            and room.access_level == old_access_level
+        ):
             return
 
-        metadata = {"configuration": room.configuration}
+        metadata = {
+            "configuration": room.configuration,
+            "access_level": room.access_level,
+        }
 
         try:
             RoomManagement().update_metadata(