You'd have to check with the landlord, and maybe make some concessions (no outside/ bathroom for example?)

ESAs are not covered under the ADA --yet-- though there are airlines that allow them IF they can fit in a carrier under the seat...but they don't have to allow them.

A letter from your doctor should be enough to get the ball rolling. You need to be able to explain a little bit as to why the cat helps you...but not too much. Even for service dogs the ADA only allows 2 questions (is that a service dog, and, can you name (3) things that the dog does for you that you can't do for yourself)... but for ESAs they can ask away (doesn't mean you need to disclose, but you might not get allowances if you don't.)

Being nice is key. Are there any renters who have service dogs? That might let landlord lean in your favor.

FYI The ADA came out and made a new "ruling"... that ONLY dogs can be service animals covered under the ADA. Yes, there are other animals that some are allowing (maybe due only to ignorance of the newest ADA notice?)---this conversation was in another thread though.

ALSO FYI there is NO national "certification" whatsoever of service animals nor ESAs. (Don't get ripped off by some online group selling you anything along those lines!) There are guidelines IF someone wishes to have their dog trained for service animal work, and some States have designators that can be used if the owner wishes (such as FL has the international orange leash/collar designated for hearing ear guide dogs) ..

May you get what you truly need.