OK at the registro civil that's what they said. Birth certificate, photocopies of both parents' passports, and immigration status. Do I have any remedy?
Immigration status is prooved just by making a photocopy of the passport with a stamp (if you don`t have DNI). I did so with a stamp, even I was legal, but nobody even tried to read it. You need to present a photocopy- you will present it, requirment is met, chin chin.
Your status now is irregular, it is a status too, many people are actually living like this for years or decades, it does not mean they don`t have the same rights like others!
Try to speak with the jefe of the office or check other office.
With other office first just try to come first with all photocopias and documents ready not for consulta but to take the date (fecha) when you want to marry, don`t just speak, choose a date, make it serious. If they deny you too, its COVID come on, tell them it is not possible to renew the visa, if you go to Chile for ex you have to cuarantine there for 14 days+make PCR (6000 pesos), say you can`t afford it, you have wife/husband, kids. dutys to do etc you can`t go out for 14 days and waste money on travel, hotel, test etc. Argentinians are usually kind people, they should understand.