♻️(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/serializers.py

@@ -2,8 +2,6 @@
 
 # pylint: disable=W0223
 
-import uuid
-
 from django.utils.translation import gettext_lazy as _
 
 from rest_framework import serializers
@@ -222,17 +220,9 @@ class RequestEntrySerializer(BaseValidationOnlySerializer):
 class ParticipantEntrySerializer(BaseValidationOnlySerializer):
     """Validate participant entry decision data."""
 
-    participant_id = serializers.CharField(required=True)
+    participant_id = serializers.UUIDField(required=True)
     allow_entry = serializers.BooleanField(required=True)
 
-    def validate_participant_id(self, value):
-        """Validate that the participant_id is a valid UUID hex string."""
-        try:
-            uuid.UUID(hex=value, version=4)
-        except (ValueError, TypeError) as e:
-            raise serializers.ValidationError("Invalid UUID hex format") from e
-        return value
-
 
 class CreationCallbackSerializer(BaseValidationOnlySerializer):
     """Validate room creation callback data."""