I can see from web searches that there are a lot of ombudsman agencies in Germany. But they are mostly private sector. I cannot find the national ombudsman that handles cases of a problem with a federal public service. Does that exist?
That’s a federalist and bureaucracy issue: depending on what issue exactly and issue it depends.
Dor example anything related to data protection even if the state the Datenschutzbeauftragte des Landes (Date protection delegate of the country/district) is the right person.
If you have something that needs escalation with the AGA then the only way I’m aware of is a form that you can send to a central complaints point.
In general the answer is “no” though sadly. The usual way is “we investigated ourselves and found no wrongdoing” from personal experience.