[Testing] Implement logging verification for web runner

Migration from https://github.com/aws/aws-durable-execution-sdk-python-testing/issues/132

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## Overview

- We have implemented a customized logger with durable execution info (ref: https://github.com/aws/aws-durable-execution-sdk-python/blob/main/src/aws_durable_execution_sdk_python/logger.py#L55).
- We should be able to verify the logging works correctly with integration tests.

## AC

- Implement a handler that checks cloudwatch logs are created correctly
- Add logging verification for logger integration tests

Start with the customized logger referenced in src/aws_durable_execution_sdk_python/logger.py and locate the existing logger integration tests. Implement the CloudWatch log-checking handler and add integration-test assertions that the durable execution logging is created correctly; done means the logging verification passes against the intended CloudWatch output.

Written by the indexing model from the issue text.