If you can show me how to reproduce this, I can maybe try to understand what is the error in ansible deployment. There might be an order in the docker daemon starting phase that needs to be respected (and yes, this is probably docker playing with iptables and adding it's NAT rules). I don't see where Ansible could be responsible but we can surely improve this.

unclear if this is still a problem to be solved.

This has not been seen for a while, we normally use ufw with Docker and it does not need any other tunning.