🛠 Tooling: integration tests don't pass locally (except for Josh 😆)

### Tooling Report Checklist

- [x] I have tried restarting my IDE and the issue persists.
- [x] I have pulled in the newest version of the project.
- [x] I have [searched for related issues](https://github.com/JoshuaKGoldberg/create-typescript-app/issues?q=is%3Aissue) and found none that matched my issue.

### Overview

Adjacent to but different from https://github.com/JoshuaKGoldberg/create-typescript-app/issues/2188. The integration tests will use the name of the repo as part of executing CTA, and for external contributors (who are running from a repo that isn't `JoshuaKGoldberg/create-typescript-app`), it fails.

Example

### Additional Info

_No response_