If you don’t trust your management, then don’t tell them. Who’d see your FMLA letter? HR? Management? FMLA has to to state the reason so it’s kind of hard to keep it a secret. Doctor has to put actual reason for your leave. It can’t be fabricated

Are you afraid if you tell your friends at work they’ll tell management? There are plenty of things we know about each other but we don’t disclose it to management if we don’t have to

You can ask your doctor to be vague. But I don’t know if it’s possible. I took FMLA few times for taking care of family members, not myself, and it was somewhat specific like what’s it for and there was no way I could hide it from management. They had to see it to give me days