Is Sri Ramakrishna Paramahamsa an avatar?
There are stories about his birth but the stories has no Scriptural Basis, it more looks like created by someone.
In the 15th century, when Sanatana Goswami (one of the six Goswamis of Vrindavana) asked Lord Chaitanya ‘How to recognize the avatars in this Age of Kali?’ Lord Replied – “His name must be there in Shastra and he must possess all the 6 Opulences mentioned in the scriptures.” If we read the full conversation of Sanatana Goswami and Lord Chaitanya we can understand that Sanatana Goswami already knew that Chaitanya Mahaprabhu is Krishna Himself but he asked that question for us or future generations. Hence we have a genuine Information from God himself to recognize avatars of him in kaliyuga.
So the Incarnation of God should be mentioned in Shastra just like there are mentions/predictions of Lord Chaitanya, Lord Venkateshvara & Lord Kalki but there is no single mention of Ramakrishna Paramhansa in any Vedic scripture.
Most importantly, if he were the incarnation of Lord, then his teachings should be Similar but he said – ‘You can surrender to anyone or you can accept any path!’ But original God Krishna, Rama, Vishnu, Kapila and other Incarnations of Supreme Lord said ‘Abandon all varieties of religion and just surrender unto Me.’
Some believers say that Ramakrishna taught what was correct according to that time, place and person. But what is the essence of his teachings? After teaching various things he ultimately said – Surrender to anyone or accept any path! Moreover, he taught that God is not a person. But Lord Krishna in Bhagavad-gita (BG 18.66 ) after speaking various principles, ultimately said, “O Arjuna, Just Surrender unto me” and in BG 14.27  he said “I be the basis of impersonal brahman”. This is the Difference between Ramakrishna and Krishna.
Therefore, all the Incarnations of Krishna appeared in different era/Yuga and taught the same teaching, i.e.”just surrender unto me”. Actually the knowledge of Gita is eternal and applicable to any Yuga or time (Ref).
Also the incarnation of Lord Shiva i.e Adi Shankaracharya in his text ‘Gita mahatmya’ Said – ‘Ekam Shastram Devki-Putra Gitam‘ means ‘There need be only one holy scripture – the divine Gita sung by Lord Shri Krishna.’
These are just a few examples. There are much Dissimilarities between the teachings of ‘Original Rama and Krishna’ & ‘Ramakrishna’. Therefore Sri Ramakrishna is not God or God incarnation.
Hare Krishna !