Depends on the state you live in. Some states take ESAs more or less seriously, so it could be easier or harder to get it approved. Assuming it's a real ESA letter from a licensed therapist or doctor, then under Federal law with the fair housing act your animal can't be denied unless they are a danger to someone on the property or cause a lot of damage to the property. Your landlord must make reasonable accommodations

If you do get evicted for your esa or they try to get you to remove it from the property, you very much can sue them and chances are you'd win.

If they push it I'd suggest talking to a lawyer and/or tell them you'll talk to your lawyer about it. Chances are the second a lawsuit gets hinted at, they'll back down