Description of problem:

Follow-up issue created while reviewing https://github.com/redhat-developer/rhdh-chart/pull/248#issuecomment-3382187902

When running the command /bump {chart-name} {bump-level}, the workflow runs through the pre-commit and updates the bump version, but does not commit the update to the PR branch if the PR is coming from a fork.
It works correctly for PRs created in the same origin repo, like the ones created by Renovate. But when the PR was created from a fork, the Workflow will push to a new branch inside the origin rhdh-chart repo, which is not what we expect.

Repro steps:

1. Create a PR against the rhdh-chart repo from your fork
2. Add a bump comment to the PR, e.g.: `/bump backstage patch`

Actual results:

• After the Bump Workflow runs and applies the pre-commit hooks, it will push the changes to a new branch in the origin rhdh-chart repo, e.g.: https://github.com/redhat-developer/rhdh-chart/tree/pullPolicy?rgh-link-date=2025-10-08T15%3A47%3A03Z

Expected results:

It should push to the same PR branch, similar to this Workflow in the Operator repo: https://github.com/redhat-developer/rhdh-operator/blob/main/.github/workflows/pr-bundle-diff-checks.yaml

Additional info (Such as Logs, Screenshots, etc):