Automatically push updated snapshots for PRs

### Describe the feature

Small CDK contributions would be much easier if a [Github action could run `cdk-integ --dry-run` and push updated snapshots](https://peterevans.dev/posts/github-actions-how-to-automate-code-formatting-in-pull-requests/), especially for PRs that are only failing those

### Use Case

I mostly send drive-by PRs and haven't been able to get CDK building in Gitpod lately:
https://github.com/aws/aws-cdk/pull/21914

Even when I had CDK building successfully in Gitpod, it took hours to spin up the environment on Gitpod, build everything, and figure out Github auth in Gitpod

Start by reviewing the repository's GitHub Actions configuration and the existing cdk-integ --dry-run workflow or entry point. Use PR 21914 and the linked GitHub Actions article as context for the failure mode. Done means pull requests can safely run the integration snapshot check and push updated snapshots when needed.

Written by the indexing model from the issue text.