It's unbelievable how stupid people are. Wouldn't it make more sense for these fragile, triggered people to have a card that says "please don't talk about my weight, that's not what i'm here for" or something like that? (still crazy, but it would address their claimed issue) Do they think doctors are blind and "oh if I don't get weighed, the doctor might think I'm 150 instead of 450" Doctors probably can't guess a morbidly obese person's weight down to the exact number, but I'm sure they can guess within 20 lbs of someone's weight. Not getting on the scale isn't hiding it.
If they don't want to find out their true weight, they can close their eyes and ask the nurse not to tell them.