I think is better to programming a lambda and obtain user and password from Secret Manager... So I think the better solution is B

I'm confused. Both A and B seem to be viable. There is no requirement of cost, complexity, or overhead. :S

https://aws.amazon.com/blogs/compute/using-api-destinations-with-amazon-eventbridge/ Amazon EventBridge enables developers to route events between AWS services, integrated software as a service (SaaS) applications, and your own applications. It can help decouple applications and produce more extensible, maintainable architectures. With the new API destinations feature, EventBridge can now integrate with services outside of AWS using REST API calls.

Answer should be B.

Configure Amazon EventBridge Scheduler to match incoming AWS Batch job SUCCEEDED events. Configure an AWS Lambda function to invoke the third-party API using a username and password. Set the Lambda function as the EventBridge rule target.

A. Configure an Amazon EventBridge rule to match incoming AWS Batch job SUCCEEDED events. Configure the third-party API as an EventBridge API destination with a username and password. Set the API destination as the EventBridge rule target. This option is the most direct and serverless approach to meeting the requirements. Amazon EventBridge can detect the successful completion of the AWS Batch job and trigger actions based on this event. By configuring the third-party API as an API destination with authentication credentials, EventBridge can directly invoke the third-party reporting application without the need for additional services. This approach minimizes complexity and operational overhead.

Create an AWS Lambda function responsible for invoking the third-party reporting application's HTTP API endpoint. The Lambda function will be triggered by the successful completion of the AWS Batch job.

AWS Batch sends job status change to EventBridge. https://docs.aws.amazon.com/batch/latest/userguide/batch_cwe_events.html

look like B