I'm sorry you are in this difficult spot, but you needn't worry. Unless you will need to have your support animal at work, there is no need to disclose anything to anybody. As long as you can perform your job competently, your boss should be happy. Most bosses are completely unconcerned with their employees' private lives, as long as that private life doesn't interfere with their work. So, to recap, as Mr. Forman notes, they cannot require you to disclose the nature of a medical condition. Indeed, there is no cause for your employer to know you are dealing with a mental health issue, unless it is interfering with the performance of your job. If you require reasonable accommodations, though, you will then be talking about being covered under the ADA. If that is or becomes the case, you will need to consult with an employment lawyer immediately.
I think you should go to work as you would have before this issue arose. Remember, nobody knows anything about your issue. Further, if they know you have an animal, as far as anyone will be able to tell, your support animal is simply a pet. You can put a picture of the animal on your desk if that helps you get through the day, as long as it doesn't violate company policy. If anybody asks "who" that is, you can honestly say, "that's my dog/cat/gecko/ferret, etc. her name is . . .".
They can't fire you for having a treatable mental health issue if you are doing your job competently. That would be a violation of the ADA. Good luck as you move forward.
We are serious lawyers for the seriously injured. I am a co-author of WEITZ ON AUTOMOBILE LITIGATION: THE NO FAULT HANDBOOK. The opinions expressed in this answer are not legal advice. These opinions are based on New York practice. We have no attorney-client relationship. Conducting a conversation with me through the avvo comments section does not create an attorney-client relationship.
Mr. Forman and Mr. Weitz give good advice. In the first instance, if your mental condition does not prevent you from performing your duties, then there is no reason to disclose your condition to begin with, particularly if you can perform your position without the phyical presence of your support pet. It is a bit more complicated if the pet has to be present on the jobsite, because that then raises questions about whether "accomodations" under the ADA are feasible (or even if your employer is covered under the ADA depending on the size of the company,,etc). If that is a possibility, get a sit down consultation with an employment discrimination attorney in your area as you do not want to post sensitive information on a public website such as AVVO.