✨(backend) add lobby cache clearing method for room and participant

Introduce new method on lobby system to clear lobby cache for specific
room and participant combinations.

Enables targeted cleanup of lobby state when participants leave or are
removed, improving cache management and preventing stale lobby entries.

src/backend/core/services/lobby.py

@@ -342,3 +342,9 @@ class LobbyService:
             return
 
         cache.delete_many(keys)
+
+    def clear_participant_cache(self, room_id: UUID, participant_id: str) -> None:
+        """Clear a given participant entry from the cache for a specific room."""
+
+        cache_key = self._get_cache_key(room_id, participant_id)
+        cache.delete(cache_key)

src/backend/core/tests/services/test_lobby.py

@@ -839,3 +839,35 @@ def test_clear_room_empty(settings, lobby_service):
     assert cache.keys(f"test-lobby_{room_id!s}_*") == []
     lobby_service.clear_room_cache(room_id)
     assert cache.keys(f"test-lobby_{room_id!s}_*") == []
+
+
+def test_clear_participant_cache(lobby_service):
+    """Test clearing a specific participant entry from cache."""
+    room_id = uuid.uuid4()
+    participant_id = "test-participant-id"
+
+    cache_key = f"{settings.LOBBY_KEY_PREFIX}_{room_id!s}_{participant_id}"
+    participant_data = {
+        "status": "waiting",
+        "username": "test-username",
+        "id": participant_id,
+        "color": "#123456",
+    }
+    cache.set(cache_key, participant_data, timeout=settings.LOBBY_WAITING_TIMEOUT)
+    assert cache.get(cache_key) is not None
+
+    lobby_service.clear_participant_cache(room_id, participant_id)
+    assert cache.get(cache_key) is None
+
+
+def test_clear_participant_cache_nonexistent(lobby_service):
+    """Test clearing a participant that doesn't exist in cache."""
+    room_id = uuid.uuid4()
+    participant_id = "nonexistent-participant"
+
+    cache_key = f"{settings.LOBBY_KEY_PREFIX}_{room_id!s}_{participant_id}"
+    assert cache.get(cache_key) is None
+
+    lobby_service.clear_participant_cache(room_id, participant_id)
+
+    assert cache.get(cache_key) is None