GITHUB_TOKEN is no longer allowed to create PRs, so use the generated
app token instead.
<!-- This is an auto-generated comment: release notes by coderabbit.ai
-->
## Summary by CodeRabbit
* **Chores**
* Improved authentication for automated sync workflows by switching to
an app-generated token, enhancing reliability of automated
documentation/troubleshooting syncs.
<!-- end of auto-generated comment: release notes by coderabbit.ai -->