♻️(backend) sync lobby and LiveKit participant UUID generation

Refactor lobby system to use consistent UUID v4 across lobby
registration and LiveKit token participant identity instead of
generating separate UUIDs.

Maintains synchronized identifiers between lobby cache and LiveKit
participants, simplifying future participant removal operations by
using the same UUID reference across both systems.

src/backend/core/api/viewsets.py

@@ -419,7 +419,8 @@ class RoomViewSet(
         try:
             lobby_service.handle_participant_entry(
                 room_id=room.id,
-                **serializer.validated_data,
+                participant_id=str(serializer.validated_data.get("participant_id")),
+                allow_entry=serializer.validated_data.get("allow_entry"),
             )
             return drf_response.Response({"message": "Participant was updated."})