automatically run integration tests in merge gate

### Describe the feature

https://github.com/aws/aws-cdk/blob/master/packages/aws-cdk/CONTRIBUTING.md#integration-tests

> Integration tests aren't run automatically since they have nontrivial requirements to run.

This is an unfair burden to place on contributors. The merge gate needs to run _all_ requisite tests.

### Use Case

The merge gate alone should be trusted. Contributors shouldn't be required to perform manual testing on top of it.

### Proposed Solution

_No response_

### Other Information

_No response_

### Acknowledgements

- [ ] I may be able to implement this feature request
- [ ] This feature might incur a breaking change

### CDK version used

n/a

### Environment details (OS name and version, etc.)

n/a

Read packages/aws-cdk/CONTRIBUTING.md at the integration-tests section to understand their nontrivial requirements. Then inspect the repository's merge-gate configuration and determine how all requisite integration tests can run automatically; done means contributors no longer need to perform manual testing beyond the merge gate.

Written by the indexing model from the issue text.