To create a custom metric for failed profile update attempts due to validation errors, the developer should use the CloudWatch PutMetricData API. This action allows for publishing of custom metrics directly to Amazon CloudWatch. Using the AWS SDK to increment a custom metric ensures that each specific scenario such as 'failed update attempts due to validation errors' is tracked accurately. While creating a new log stream and logging error messages to CloudWatch Logs does provide persistence of error data, it does not fulfill the requirement of creating and utilizing a custom metric. Similarly, using AWS X-Ray to trace requests would provide insights into the request flow and latency but would not create a custom metric. Finally, emitting logs with the embedded metric format (EMF) would be a valid approach for collecting custom metrics from Lambda logs, but since the question specifies using the PutMetricData API, the correct approach is to use the AWS SDK to call PutMetricData.