ok I thought it would be a better mood to have an actual fax maschine, but between the extra, unnecessary complications from the unexpected death of real telephony in favour of VoIP, and the maschine I got not actually turning on, I decided to give up and just pay for an online fax service.
which promptly put me into a manual request to verify my documents before they let me have a number, and Gods only know if they'll take my DGTI.
so my best hope is to take time off work tomorrow and send a letter to the Sozialgericht (mit Einschreibung, natürlich) with a copy of the letter of the Einbürgerungsbüro, as soon as I can so that there's a chance it's delivered and inboxed still tomorrow. after all, they took their sweet time to respond to my original letter (20 days). I have 30 days and 53 minutes left in Germany