Sorry to hear that Fiscal. The letter in question issued by the AR consulate (costs 40 USD) is not sufficient to enter on its own. The only guaranteed entry is an actual visa (issued after Match 16) or residencia vigente (or one that "would have expired" after March 16) or en tramite (aka Precaria). I was issued the same letter as you back in September before I traveled to AR on an expired DNI (which had been legally extended by migraciones due to Covid). The letter or "certificado" as they call it -- was actually (in my mind) meant for the airline staff at the check in counter, to convince them I am good to go. At migraciones at EZE they did not even look at the letter/certificado. It's a "secondary" respaldo of residence that needs to be backed by a primary document.
You should never trust Argentine authorities. And always double and triple check, confirm and have a plan B. I remember that I spoke with the jefa de del consulado and she advised me before issuing the letter (which by the way has no expiration date!) that it's ultimately up to the immigration officials in Argentina to determine if you are to be admitted or not.
Your next step is to get that birth certificate and get an actual visa from the embassy in Lima. Or wait till Argentina opens the borders (likely) next month. I do not believe the document you have signed have any implications for re-enter.
Again, sorry to hear about this horrible situation. Hope you at least have a safe environment in Lima.