I found the problem. The installation is certainly not intended this way, but I was able to solve it this way.
The steps I described in point 4 of my problem description must also be carried out for the Docker container “frappe_docker-frontend-1.” In other words, for the frontend.
Here are the commands.
docker exec -it frappe_docker-frontend-1 /bin/bash
bench get-app --branch main print_designer
bench --site frontend install-app print_designer
bench --site frontend migrate
docker restart $(docker ps -q)