I am sorry to hear about your children. We will pray for them.
What you describe are psychiatric issues, to which there is no indication from your narrative of any demonic involvement. The psychiatric approach needs to be continued, but since demons like to hitchhike on problems, you can pray a Hedge Prayer of Protection around your household, and particularly them. This prayer in found in our Spiritual Warfare Prayer Catalog, linked below.
As for hypnosis, I would ask if there is any other therapy possible. Hypnosis can be dangerous spiritually and psychologically.
After reading this you will need you use your judgment, with prayer, as to the efficacy of hypnosis in your child's case.
The Use of Hypnotism
Hypnotism consists of a "state of mental absorption in which all distracting thoughts are for the moment warded off, and only such thoughts as are suggested by the hypnotist reach the consciousness of the patient. The essence of hypnotism is the concentration of mind on one idea or only a few ideas dictated by the hypnotist."[1] If the hypnotist is not skilled in this practice, the patient may suffer various evil effects.
Among the deleterious effects that may be produced are
(1) an abnormal proneness to become drowsy and to fall asleep;
(2) a detachment, more or less pronounced, from reality, even though the patient is to all appearances in a state of wakefulness;
(3) an occasional loss of memory and of mental equilibrium resulting in increased irritability;
(4) incoherence of speech. The hypnotist has the subject in his power and can strongly influence his mind. He is able, moreover, to force him to reveal secret knowledge, to fill his memory with unwholesome suggestions, and even to perpetrate immoral actions with him.
There is of course nothing wrong in itself with the practice of hypnotism, and therefore under certain conditions its use would be licit. Since hypnotism, however, violently deprives the subject of the full use of reason and free will, a justifying cause is required for allowing it to be practiced. Hypnotism may be licitly used provided three conditions are verified:
(1) there is present a grave reason,
(2) the consent of the subject is obtained, and
(3) due precautions are observed.
We shall explain in detail the meaning of these three conditions.
A Grave Reason.A proportionate cause is required in order that one licitly be hypnotized, for we are not permitted to give up, without a compensatory reason, our dominion over the faculties of the understanding and the will. A justifying reason would be, for example, the need to cure or to curtail an evil habit such as drunkenness, pyromania, masturbation, or kleptomania. If deemed medically advisable, hypnotism may be licitly employed in place of an anesthetic for surgical cases. If there is at hand another remedy which would be equally effective but would not involve the dangers that often accompany hypnotism, it must of course be preferred.
The Consent of the Subject. The consent of the patient must be procured, for no one has the right to deprive another, against his wishes, of the full use of his faculties. Hence forcibly to impose an hypnotic state on another, even for a short period, would be to violate his rights. It is not necessary, however, always to obtain the explicit consent of the patient. If the patient is in such circumstances that his explicit consent cannot prudently be sought and if one is convinced that he would not object to hypnotic treatment, this would be considered sufficient consent on the part of the patient. With regard to the insane and to children who have not reached the age of reason, the physician should not use hypnotism without first procuring the consent of the parents or of the one who is charged with their care.
Due Precautions.The first precaution which must be observed is that the hypnotist be one who is medically qualified to exercise this art. An unskillful hypnotist may injure the patient's mental faculties. A competent physician can as a rule prevent the evil effects which sometimes result from the use of hypnosis.
The second requirement is that there be present an authorized witness of unimpeachable character who will serve as a protection both to the physician and to the patient. The witness (for example, a parent or the marriage partner of the patient) could afterward defend the physician against any false accusations of improper conduct.[2]
ENDNOTES
1 James J. Walsh, M.D., , p. 152. New York: D. Appleton and Company, 1929.
2 For material on hypnotism and its morality see James J. Walsh, M.D., Psychotherapy, pp. 151-62 (New York: D. Appleton and Company, 1929); Catholic Encyclopedia, article "Hypnotism"; William T. Heron, Clinical Applications of Suggestion and Hypnosis, chap. 8, pp. 90-93 (Springfield: Charles C. Thomas, 1950).
(The following excerpt is taken from "The Modern Catholic Dictionary" by Fr. John A. Hardon, published by Doubleday & Company Inc.)
HYPNOTISM. The phenomenon of artificially induced sleep, which renders the victim abnormally open to suggestion. The subject of hypnosis tends to be dominated by the ideas and suggestions of the hypnotist while under the induced spell and later on.
According to Catholic principles, hypnotism is not wrong in itself, so that its use under certain circumstances is permissible. But since it deprives the subject of the full use of reason and free will, a justifying cause is necessary for allowing it to be practiced. Moreover, because hypnotism puts the subject's will in the power of the hypnotist, certain precautions are necessary to safeguard the subject's virtue, and to protect him or her and others against the danger of being guilty of any injurious actions. For grave reasons, e.g., to cure a drunkard or one with a suicide complex, it is licit to exercise hypnotism, given the precaution that it is done in the presence of a trustworthy witness by a competent and upright hypnotist. The consent, at least presumed, of the subject
must also be had. Several documents of the Holy See set down the norms to be
followed in the use of hypnotism (The Holy Office, August 4, 1956; July 26, 1899).