Run brew bump-formula-pr with GitHub App Token

[5ouma]

Description of issue

I use brew bump-formula-pr to keep formulae up-to-date via dawidd6/action-homebrew-bump-formula.
I used PAT with public_repo permission, and it worked. I wanted to use GitHub App Token to change the PR opener to a bot. However, it doesn't work with this error even when I added any permissions.

/home/linuxbrew/.linuxbrew/Homebrew/Library/Homebrew/utils/github/api.rb:349:in `raise_error': GitHub API Error: Resource not accessible by integration (GitHub::API::AuthenticationFailedError)
HOMEBREW_GITHUB_API_TOKEN may be invalid or expired; check:
  \e[4mhttps://github.com/settings/tokens\e[24m
	from /home/linuxbrew/.linuxbrew/Homebrew/Library/Homebrew/utils/github/api.rb:272:in `open_rest'
	from /home/runner/work/_actions/dawidd6/action-homebrew-bump-formula/v3/main.rb:[59](https://github.com/5ouma/homebrew-formula/actions/runs/7895707445/job/21548603535#step:7:63):in `<module:Homebrew>'
	from /home/runner/work/_actions/dawidd6/action-homebrew-bump-formula/v3/main.rb:17:in `<main>'

Do you have any ideas to solve this, or it's impossible to use GitHub App Token?

Related: #4514

https://github.com/5ouma/homebrew-formula/actions/runs/7895707445/job/21548603535