One more authorizers question: with lambda authorizers - is the somehow response cached? I changed authorizer response, SST says - deployed, but still getting old response.

It defaults to a 5 minute cache. You can edit it with resultsCacheTtl. See: https://docs.aws.amazon.com/cdk/api/latest/docs/@aws-cdk_aws-apigatewayv2-authorizers.HttpLambdaAuthorizerProps.html

Thanks again, Garret! I owe you a beer!