Update for you guys:
My girlfriend decided to hold back on a lawyer, we looked at the court records for the management company, the owners of the unsold shares (Aka the unit), and the company that runs the co-op only have court records for about once per year.
From this information, we drafted a letter that contained all the information that the DHCR customer service rep told us to include. (Claim of succession rights, lived here for 3 years with GMA, want all future renewals placed in her name, respond to us at this address) and we included it with the check payment for this month. It was sent with a certificate of mailing (because they refuse to collect their certified mail, for whatever reason.)
It is with a slight sigh of relief that they cashed the check yesterday, I take this as a good sign? this is not an indication that all is well but we take this as a good sign and them realizing that we do have legal grounds to be here and to fight it might not be the best decision on their end.
Is there any more pertinent information or things we should know from here? I know after searching around on this forum other landlords will refuse to cash the checks and I'm just happy that ours did not.
Oh, I almost forgot we also wrote "(UNIT #) For April 2022 Rent Only" on the check upon the recommendation of this forum. So we will also continue doing this and keeping records of everything, I am also keeping the envelopes their letters come in as per the recommendation of this forum.
Thank you so much for your help so far! I will keep updating this as we gain more insight and hopefully also help someone down the road.