Is it permissible for children to secretly transfer the ownership of their mentally ill father's real estate to themselves, using a general power of attorney to preserve his wealth and prevent it from being squandered, given that he refuses treatment and mismanages his money?

You are not permitted to transfer ownership of your father's properties without his permission. If he is managing his money unwisely, advise him gently and involve someone who can counsel him. If that does not work and he continues to act foolishly, refer the matter to the judiciary to have a legal interdiction placed on him; there is no disobedience in that.