Run checks in automatic update PRs

[fricklerhandwerk]

Technically the checks did run, the run is this one, but the run isn't linked to this PR so the checks aren't visible in the UI.

This happens for two reasons:

1. The checks are triggered by this action, which is a workflow_dispatch and workflow dispatches are considered a "manual run", so they don't update PR information. Only ["pull_request", "pull_request_review", "pull_request_review_comment", "pull_request_target", "push"] actions can link and update an action run to a PR.
2. The reason checks aren't triggered by default is this issue. The tl;dr is that GitHub doesn't allow (most) actions to create further actions when using the default GITHUB_TOKEN. There are various workarounds we can use, I think going with the last one (GH App token) makes the most sense in our case.

Originally posted by @Erethon in #329 (comment)