I do not have in my possession The Sermon on the Mount According to the Vedanta which is the book that you are most likely referring to, but I have the original articles (as part of another book) that it was based on. Assuming that few if any changes were made to the content, it is safe to say that this work has very little to do with Christian doctrine and so cannot legitimately be considered as a work of comparative religion. Rather, Swami Prabhavananda reflects upon the teachings of the Sermon in themselves, with little or no reference to other Christian writings or teachings, and applies their universal content to the context of his own experiences within the Hindu tradition. I do not recall encountering anything problematic in this book or in The Spiritual Heritage of India and would recommend the former as both interesting and inspiring and the latter as an excellent informative resource. I have, however, occasionally encountered certain problematic ideas in his work and it might be well to explain some of them in case you do happen to encounter them in either of these books. In order to do this, it is first necessary to say a few words about the organization of which he was a representative.
Swami Prabhavananda belonged to a monastic society called the Ramakrishna Order of monks, named after the teacher of the original monks who founded it, Sri Ramakrishna Paramahamsa. He (Ramakrishna) was a great spiritual master of the 19th century who combined within himself the qualities of bhakti and gnani, or devotional worship and realization. The essence of his teaching has been aptly designated as a Vedanta-Japa synthesis. A small group of young Hindu men constituted his immediate disciples and it was primarily to them that he directed his teachings but he was known and loved by people of many religions and social groups. He taught his disciples to worship God as they were accustomed, according to the traditional rituals and prayers to various deities, but also to realize his nature as the transcendent Brahman, or Absolute Reality of which these various deities were as so many contingent manifestations. To achieve this, he taught that Japa, or the repetition of God’s Name, was the spiritual method most suited to the conditions of this age, the Kali Yuga. He himself was a devotee of the great Goddess Kali and lived for many years at the compound of her temple at Dakshineswar where he presided as priest.
There are many interesting anecdotes to tell of his personality and teachings, and for these I would recommend to you The Gospel of Sri Ramakrishna particularly in its abridged form (because of its immediate accessibility to the general reader), a book that I think all people who are endowed with a love of God should be exposed to regardless of their religious background. The most important idea that I would like to convey is that Ramakrishna was a very simple man and without formal education, who followed the traditional prayers and rites of orthodox Hinduism. He was guided in the path of advaita from a wandering monk and later in tantric rites by a female priest. Through their training and his innate spiritual gifts he attained to the realization of his identity with the Absolute and was forever prone to frequent bouts of ecstasy when this realization would return to him. In fact, it is sometimes said (I believe originally by he himself) that after experiencing this realization, his normal state of consciousness became that of samadhi, characterized by ecstatic absorption into the Absolute, and it was only through an effort of concentration that he was able to maintain a level of consciousness such that he might communicate with his disciples and function within the phenomenal world. He taught to his closest disciples some of the spiritual methods that he had learned, and conveyed to them a transmission of spiritual influence that initiated them into the spiritual life.
Following the master’s death, and after a period of wandering, these disciples took monastic names and vows and organized themselves into the Ramakrishna Order of Monks. The two most prominent disciples were Swami Brahmananda and Swami Vivekananda. The former was a soft spoken and pious Hindu whom Sri Ramakrishna considered to be his spiritual son. He had come to live with Sri Ramakrishna and learn from his words and presence the meaning of the spiritual life. Swami Vivenakanda was an outspoken and sometimes controversial member of the Brahmo Samaj, a Hindu reform movement. Sri Ramakrishna loved him dearly and viewed him as the acknowledged leader of the group due to his great mental prowess, oratory and rhetorical skill. He would frequently request Vivekananda to visit him so that, like Swami Brahmananda, he might receive the benefit of his spiritual influence and guidance, and thereby make the best use of his innate talents. By Vivekananda’s own admission, “He was afraid about me, that I might create a sect if left to myself.” The nature of these two monks represents the contrary influences that inform and characterize the present movement. On the one hand there is the heritage of orthodox Hinduism and the faithful adherence to the teachings of Ramakrishna and on the other, the modern ideologies deriving through Vivekananda from the Brahmo Samaj.
As a disciple of Swami Brahmanada, Swami Prabhavananda is characterized generally by orthodoxy and faithfulness to Ramakrishna as to his master, but the pervasiveness of Vivekananda’s personality and teachings within the Order are such that he was unable to escape their influence. Certain points of contention introduced by Vivekananda include the preoccupation with social welfare, a perspective concerning meditation as a scientifically justified and quantifiable method of realization, and the view that the modern world is the result of progress in the material realm which must ideally be synthesized with the progress of Indian civilization in spiritual realm. Swami Prabhavananda does not necessarily prescribe to these views in their entirety, and to my recollection they may be entirely absent from the works that I recommended. Nevertheless, one can occasionally catch a glimpse of Vivekananda’s influence at work in his more popular articles and it is best to be aware of such ideas such that we do not fall prey to them inadvertently by accepting them alongside those that are not objectionable.
In his very informative book Introduction to the Study of the Hindu Doctrines, Rene Guenon provides some background on the Brahmo Samaj and additionally, the following remarks on the Vedanta movement deriving from the Ramakrishna Order. He wrote,
“Another still more completely aberrant branch, better known in the West, is that founded by Vivekananda, the disciple of the illustrious Ramakrishna though unfaithful to his teaching; it has recruited its adherents mostly in America and Australia, where it runs ‘missions’ and ‘temples.’ There Vedanta has become, like Schopenhauer’s conception of it, a sentimental and ‘consoling’ religion, with a strong dose of Protestant ‘moralism’; in this degenerate form, it approaches very close to Theosophism, towards which it stands in the position of natural ally rather than a rival or competitor. The ‘evengelical’ attitude assumed by this pseudo-religion has earned it a certain success, chiefly in Anglo-Saxon countries; while its inherently sentimental character is well attested by the ardour for propaganda animating its votaries; for, as might be expected, an altogether Western propensity for proselytism rages intensely in these organizations, which are Eastern in nothing but the name, apart from a few merely outward signs, calculated to interest the curious and to attract dilettantes by playing on their taste for an exoticism of the feeblest type. This so-called Vedanta, which is a product of that queer American and characteristically Protestant creation called the ‘Parliament of Religions, ’ and which pleases the West all the better the more completely it is distorted, has practically nothing left in common with the metaphysical doctrine the name of which it bears. No more time need be wasted on it; but it seemed best at least to mention its existence, in order to put people who have heard of it on their guard against possible false assimilations; as for those who have not come across these movements, it is best that they should be made aware of them, since they are not nearly so harmless as might appear at first sight.”
I think that, contrary to Guenon’s assertion, a few more words might be said concerning the orthodox presence within the Society deriving from Ramakrishna’s spiritual transmission as well as the possibility of a westerner practicing Hinduism and the Path of Gnana Yoga in an authentic manner. I would be willing to devote a few more words to these subjects, but for now I will simply say that if you would like to learn about Hinduism or Christianity it might be best simply to study the sacred texts and their orthodox commentaries, avoiding altogether that which, despite whatever merits it might possess, may give rise to confusion or misinterpretation.