Wa-Alaykum As Salaam
Not familiar in particular with this form of medicine, however according to University of Minnesota, "Ayurveda is considered a complementary healthcare option, with many Americans employing Ayurvedic elements such as massage, meditation, or cleansing therapies.". It is considered the oldest form of medicine!
That's not quite true though (being the oldest form of medicine, as Prophet Adam peace be upon him, was sent down to earth with "the science of the religion and the science of the body", indicating that health and medicinal knowledge was sent down with him and so medicine of the prophets may be considered the oldest form of medicine.
Back to your question - is it halal?! A number of issues need to be resolved:
1. Firstly, spiritually, is it based on the belief that Allahï·» is the Healer of all ills? If other than Allah» is sought, then of course it is considered not halal.
2. Medicines used need to not contain anything forbidden. Medicine needs to be evidence based - either in the scientific sense (repetitive results achieved) or in the sense of religious (evidence the Imams and Prophets used to treat ailments in this way). So you may be granted permission o use something that contains a non halal product if you are in need of it medically, however, that need needs to be reflected in evidence.
3. For example - vomiting and enema is considered a form of prevention and treatment in Islamic way of life and health also, as well as herbal medicines. Massage is very important (narrations say do not be surprised if the dead come back to life with massage!) and steam baths were common in the days of the sunnah of the Prophet, peace be upon him and his Purified Household. So these treatments are not contrary to Islam.
4. If the doctor will be using the name of Allah» alone - is the doctor muslim or only using the name Allahï·» to please you, and in fact the doctor has no faith in Allahï·» what so ever and is calling upon another false god.
As you stated, they use what is in sacred texts of a non islamic origin so the initiation of that medicine is questionable.
With regards to references, please see here <http://practicalislam.online/2017/07/04/islamic-medicine-introduction-part-1/> for introduction to a form of alternative medicine closer to what you may be looking for.
We pray and hope for your quick recovery from whatever illness and that you find a suitable treatment.