You can invoke a Lambda function asynchronously via API Gateway only if the integration is non-proxy. By default, HTTP APIs are designed to support only proxy integrations for Lambda and HTTP endpoints so it is not possible to set the X-Amz-Invocation-Type header in the API Gateway integration config.

In order to invoke a Lambda asynchronously via HTTP API, you can use two Lambda functions with one acting as proxy to your actual function.

HTTP API --> Invoke Lambda1 synchronously --> Invoke Lambda2 asynchronously

Below is a sample NodeJS code snippet for Lambda1 to invoke Lambda2 asynchronously -

const params = {
        FunctionName: 'FUNCTION_NAME',
        InvocationType: 'Event',
        Payload: JSON.parse(event.body) // this is the event coming from API Gateway to Lambda1
    };
await lambda.invoke(params).promise(); // await here is only going to wait for the HTTP request to be successful. Once Lambda2 is invoked, it will return immediately