>I know about FMLA leaves of absence, but that requires you (!) to disclose your illness to your employee; given the massive stigma around mental illness, I really, really dislike that option.
Not true.
You don't have to disclose the nature of the condition, simply that a serious condition exists.
It doesn't have to specify what the condition is, just something demonstrating that a physician has validated your need to take leave for a condition protected by FMLA.
In any case, providing a medical certification only involves the HR department of an employer and there are clear rules about non-disclosure to people the applicant works for/with.
Source: Personal experience with the process and extensive conversation with people who deal with such things for a living.
Not true.
You don't have to disclose the nature of the condition, simply that a serious condition exists.