I am not sure how things in the USA, but this seems it would work in fairytale land. In reality, you might get fired over a made-up reasons, because there are too many people who don't need accomodation. If you can tough it up, somehow, I think it is better not to disclose.

I actually looked up the ADA because my mother the labor organizer showed uncharacteristic restraint and muttered something about discrimination (normally she gets loud and righteous about this kind of thing). Once the Dean of students got wind of the situation everyone suddenly got worried that I might sue them (which I easily could have). An employer can give another reason for firing a person but the disability element would still leave them open to a possible lawsuit and/or depending on their visibility level, damage the reputation of the company.