Does anyone can help me with the issue I described above?

Can you show network requests when you hit the API? Is there a preflight check?

Where do I put this in my code?

const processor = new sst.Function(this, "Processor", "processor.main");
new SubscriptionFilter(this, "Subscription", {
  logGroup,
  destination: new LogsDestinations.LambdaDestination(processor),
  filterPattern: FilterPattern.booleanValue("$.error", true),
});

You can enable access logs using

accessLog: true

in the Api construct’s props.