You cannot configure API Gateway with a static IP address. Instead, the method you describe (Lambda, VPC, NAT Gateway) is the way to achieve this.
Instead of using a static IP for API Gateway when it is calling the backend, you may want to consider using this - https://docs.aws.amazon.com/apigateway/latest/developerguide/getting-started-client-side-ssl-authentication.html
As mentioned in this document - "You can use API Gateway to generate an SSL certificate and then use its public key in the backend to verify that HTTP requests to your backend system are from API Gateway. This allows your HTTP backend to control and accept only requests that originate from Amazon API Gateway, even if the backend is publicly accessible."
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