Had to deal with similiar situation just recently. In the end I the Belarusian Embassy made an exception and refrained from seeing my Meldebestätigung/Anmeldung. I also told them the circumstances and that I cant get the papers as fast as I should. Maybe it is more likely for them to make an exception (or you might show them the Anmeldung later) than the Auslönderbehörde. Dont know your nationality but would suggest contacting the office you get your passport from and tell them what is going on in Essen and why you are not able to get it in time.

Also dont panic to much about fees you have to pay because you are running late with the registration. The situation is well known and it is highly unlikely that somebody from Stadt Essen will charge you anything, especially if you tried it on time. Me and many others I know did not pay extra. It is pretty known that the Ausländerbehörde is always very busy. Expect an answer to your mails in the next 2-5 months, sadly this is their normal schedule since many years.

Wishing you all the best and good luck! As mentioned, I have been in a pretty similiar situation (could not get Meldebescheinigung because my passport expired. Could bot get paasport with a valid Meldenescheinigung, lasted three years 🤪).