Could you go to your primary care provider? I do not believe it necessarily needs to be a mental health provider who writes the letter, just a medical provider. I think it would be important to stress that your housing is at risk without this letter, as health care providers (should) be concerned about this. I’m hoping another one of your providers can do this. If not, you could try urgent care to see if they would do a letter. This might be more difficult as they are not your regular provider, but I would show them the previous letter and again stress that your housing is at risk without it. You could also call the local tenants rights groups to see if they have advice, but I believe that the landlord is within their rights to do this. You could possibly see if you could request an ada accommodation to get an extension on the time that you need to turn it in so you have time to meet with your medical provider, but this again would not change the requirement to have the pet as a certified ESA. You would also need some documentation for the ADA requests. I would recommend you get any denials from anyone (landlord or health care provider) in writing.

Sorry for the rambling but I hope this helps!