♻️(all) stop relying on cookies for the lobby flow

The lobby system relied on cookies to identify the participant
across the wait/enter cycle, which does not work in an iframe
context where our cookies are dropped.

Simplify the lobby behavior:

* The POST request that enters the lobby now returns the
  participant id in the response.
* The frontend passes that id back on subsequent requests to keep a
  sticky session while trying to enter the room.

This moves a bit more logic to the frontend but should be a
transparent refactoring, without decreasing the security of the
lobby flow.

src/backend/core/api/throttling.py

@@ -1,11 +1,11 @@
 """Throttling modules for the API."""
 
-from django.conf import settings
-
 from lasuite.drf.throttling import MonitoredThrottleMixin
 from rest_framework.throttling import AnonRateThrottle, UserRateThrottle
 from sentry_sdk import capture_message
 
+from . import serializers
+
 
 def sentry_monitoring_throttle_failure(message):
     """Log when a failure occurs to detect rate limiting issues."""
@@ -42,13 +42,14 @@ class RequestEntryAnonRateThrottle(MonitoredAnonRateThrottle):
     def get_cache_key(self, request, view):
         """Use the lobby participant cookie ID as the throttle cache key.
 
-        Only throttle if a cookie is already set. If no cookie exists yet,
-        return None to skip throttling — the cookie will be set on the first
-        response, and throttling will apply from the second request onward.
+        Only throttle requests carrying a participant identifier. The
+        identifier is returned by the first request-entry response and
+        echoed back by the client from the second request onward, which is
+        when throttling starts applying.
 
-        Keying on the cookie rather than the IP address prevents penalising
-        multiple users behind the same NAT/proxy, and is consistent with how
-        LobbyService identifies participants.
+        Keying on the identifier rather than the IP address prevents
+        penalising multiple users behind the same NAT/proxy, and is
+        consistent with how the lobby identifies participants.
 
         Note: as per DRF documentation, application-level throttling is not a
         security measure against brute-force or DoS attacks. This throttle exists
@@ -58,10 +59,14 @@ class RequestEntryAnonRateThrottle(MonitoredAnonRateThrottle):
         if request.user and request.user.is_authenticated:
             return None  # Only throttle unauthenticated requests.
 
-        participant_id = request.COOKIES.get(settings.LOBBY_COOKIE_NAME)
+        serializer = serializers.RequestEntrySerializer(data=request.data)
+        if not serializer.is_valid():
+            return None
 
-        if participant_id is None:
-            return None  # No throttling for cookieless requests
+        participant_id = serializer.validated_data.get("participant_id")
+
+        if not participant_id:
+            return None  # No throttling for unidentified requests
 
         return self.cache_format % {
             "scope": self.scope,