How to fill out and apply for Subject Access Request (SAR)? is for the UK, how do I do this for Hungary?
The Hungarian National Authority for Data Protection and Freedom of Information has a form in Hungarian which requests data handled and everything related from whoever is handling that data.
In the long stay process PDF says
If requested by the applicant, the consulate and the Office for Immigration and Nationality within its jurisdiction shall provide information regarding his/her data on record, unless the Aliens Act prohibits the disclosure of such data.
This also happens to be in the privacy act.
After that, you could lodge a complaint with the Data Protection Authority again in 1. and theoretically they might enforce a ruling but I am not holding my breath against one part of the Hungarian government ruling against another part. That'd be a surprise.