The best solution that meets the requirements with the least operational overhead is Amazon API Gateway with provisioned Lambda concurrency. This option allows for scalable and serverless operation, meaning you don't have to manage any servers or containers. Provisioned concurrency ensures that your Lambda functions are pre-warmed and can respond immediately to requests, reducing latency. This setup provides high availability, automatic scaling, and low operational overhead, making it ideal for handling a large number of API requests consistently and quickly.