Is the authenticity of "the Prayer of Intimacy in the Grave" and the hadith mentioned therein narrated from the Prophet, peace and blessings be upon him, and to what extent is its attribution to the book "Fath al-I'lam" in Shafi'i jurisprudence authentic?

The prayer mentioned after burying the deceased is not from the guidance of the Prophet, peace and blessings be upon him. All that is authentically reported from him is supplicating for the deceased, such as his saying: "Ask forgiveness for your brother and ask for him steadfastness, for indeed he is now being questioned." Designating a specific prayer with a specific recitation for the deceased is an additional innovation (bid'ah). The hadith mentioned in the question is not found in the Sunnah collections, and the signs of fabrication are evident upon it due to the promise of a great reward for a small action.