Employ AWS CDK for deploying API Gateway and Lambda functions, updating the AWS CloudFormation stack for new code deployments, and utilizing a Route 53 failover routing policy for the canary release strategy.

Utilize AWS CloudFormation to deploy API Gateway and Lambda functions with Lambda function versions, updating the CloudFormation stack with new Lambda code, and employing a canary release strategy for API version updates, with promotion of the new version post-testing.

Deploy API Gateway and Lambda functions using AWS Elastic Beanstalk, deploying new API and Lambda function versions for code changes, and gradually shifting traffic with an Elastic Beanstalk blue/green deployment.

Deploy API Gateway in the service layer and Lambda functions in a custom layer using AWS OpsWorks, performing a blue/green deployment for code changes and gradually shifting traffic with OpsWorks.