Utilize API Gateway to input requests directly into DynamoDB, employ a DynamoDB stream for invoking the third-party API, and eliminate the Lambda function.

Direct API Gateway requests to an Amazon SQS queue, set Lambda's reserved concurrency to the third-party API's threshold, and have Lambda process messages from the SQS queue.

Send API Gateway requests to an Amazon SNS topic, set Lambda's provisioned concurrency to match the third-party API's threshold, and process messages from the SNS topic.

Input API Gateway requests directly into Amazon Athena, use SQL for data transformation with query results directed to both DynamoDB and the third-party API, and remove the Lambda function.