Sorry to hear about your additional paperwork problems...
Unfortunately, I have no residence in Hungary so this option is not available.
I don't have a residence in Hungary either, but I do have a Hungarian passport, so you should be able to get yours too... In the worst case scenario, when we renewed our passports the last time, it lasted maybe 2-3 weeks... and we did it in the "old-style", going to the office, etc...
Good luck with the paperwork and taxes, and hopefully, see you in Mexico!
Yeah I think it moght be the easiest way to just get an appointment in Hungary.
Thanks for the good wishes!