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Just in case anybody is wandering what was needed was to add as the message said the lambda permissions using something like:

// Give permissions from API Gateway to invoke the Lambda
let invokePermission = new aws.lambda.Permission("api-lambda-permission", {
    action: "lambda:invokeFunction",
    function: lambda,
    principal: "<http://apigateway.amazonaws.com|apigateway.amazonaws.com>",
    sourceArn: deployment.executionArn.apply(arn => arn + "*/*"),
});