Absolutely you should speak.
If I was let out of my “prison” occasionally, I might not be here now.
Never be embarrassed to tell the truth. Don’t worry about other people’s opinions. If your son had a broken leg, would you be concerned about showing his leg in a cast? Mental health is the same. If people care positively about your family, why keep them away? If people care negatively, it is none of their business.
How is disclosing anything about your health to a medical practitioner “not fair”. Your son knows the issues already, how is hiding him in a “prison” going to help him?
I know several people who refuse to disclose information about themselves and their families, relying on Dr Google and “heresay”. They don’t get better. They become more withdrawn.
My 2c... sorry, you hit a sore spot for me. I think that keeping secrets about your son would be a mistake. He will resent you for doing that later in life. Like I do with parents. And then there will be guilt at the resentment. There are no upsides in hiding.
Have you asked your son what he would like to do?