Is it permissible to resort to a Quranic healer over the phone who uses the patient's name and his mother's name (which is a sign of a sorcerer), under the pretext that "necessities permit the forbidden" and that the patient's brother, whose condition is worsening and who is not responding to treatment with ruqyah or doctors, is "deprived of his will" and is being invited to Christianity?

The entry of jinn into the human body is established by the consensus of the imams of the Sunnah. If your brother has lost his mind due to jinn possession or other causes, then he is among those for whom the pen has been lifted, so there is no concern about his refusal to pray. Ruqyah al-Shari'ah is what has been permitted by the Sacred Law, and one should suffice with it, along with supplicating for him, and staying away from sorcerers and charlatans.