Bypass branch protection with action and Github app - github

Question

I have an action, that automatically indexes all files in the repository and creates a csv. Currently this action always creates their own pull request. This creates the annoying need to approve two pull requests per change (the first one with the change itself and the second one with the change in the index.csv file created by the action minutes later).
What I tried now, is creating a Github App, which is added to the "Allow specified actors to bypass required pull requests" in the branch protection and using tibdex/github-app-token#v1 to create a token in the context of the GitHub app.
Unfortunately, it still does not work. I get the following error:
Run echo "Hello World" >> HelloWorld.txt
[main f8445ab] Add unncessary file for testing
1 file changed, 1 insertion(+)
create mode 100644 HelloWorld.txt
remote: error: GH006: Protected branch update failed for refs/heads/main.
remote: error: Required status check "*****" is expected. At least 1 approving review is required by reviewers with write access.
To https://github.com/***/***.git
! [remote rejected] main -> main (protected branch hook declined)
error: failed to push some refs to 'https://github.com/***/***.git'
Error: Process completed with exit code 1.
The code I'm trying to run in the action is the following:
echo "Hello World" >> HelloWorld.txt
git config --global user.email "****"
git config --global user.name "****"
git add .
git commit -m "Add unncessary file for testing"
git push "https://action-name:${{ steps.generate_token.outputs.token }}#github.com/***/***.git"
I know that it is bad practice and breaks the permission concept to be able to do this, but in this case, it is fine for many reasons which go beyond the discussion at this point.
I'm happy for any suggestions on how to achieve this.
Thanks so much in advance