As a physical event that's supposed to have taken place, a splitting and reassemblage of the moon, we have absolutely no physical evidence in support of this. We are also missing a historical record for this from any non-Muslim sources.
Moreover, while there is a tradition that says Mohammed split the moon in two and further that there may have been distant witnesses about that, it looks to be quite a stretch from the basis of the Koranic verse, Sura 54:
The Hour has come near, and the moon has split [in two]. And if they see a miracle, they turn away and say, "Passing magic." And they denied and followed their inclinations. But for every matter is a [time of] settlement.
That means as I read it to be an apocalyptic vision, not a documented event. The 'later traditions' that refer to that event as not in the future but in the past are then first found in hadiths.
One of its chapters, named “The Moon,” describes the splitting of the moon as an event that precedes Judgment day (Q 54), but this event was later claimed to be a miracle performed by Muhammad (d. 632) in proving his prophethood to the unbelievers of Mecca.
–– Juan E. Campo (Ed): "Encyclopedia of Islam" Facts on File: New York, 2009.
Nobody has to take my word for it, though. Muslim and non-Muslim exegetes offer a multitude of understandings for this passge:
In classical exegesis, Q 54:1 (iqtarabati l-sāʿatu wa-nshaqqa l-qamaru) is mostly interpreted as referring to the splitting of the moon, a miracle allegedly granted to Muḥammad, but usually the classical exegetes also discuss the — equally possible — eschatological and metaphorical readings of the verse. In contrast, modern exegetes mostly confine themselves to discussing only one interpretation, but they differ radically in their conclusions and thus add a number of new—and sometimes rather bizarre—interpretations to those known from the classical tradition. While some exegetes try to minimize the miraculous aspect of the verse and offer alternative readings—historical, eschatological, metaphorical, or symbolical—others explicitly defend a miraculous reading of the text and try to adduce new arguments for this interpretation. The article draws attention to regional and confessional differences in the interpretation and shows the importance of non-scholarly exegetes and the Internet in assessing how verses from the Qurʾān are understood by Muslims today.
–– Andreas Goerke: "Die Spaltung des Mondes in der modernen Koranexegese und im Internet", Die Welt des Islams, vol. 50, no. 1, pp. 60-116, 2010. DOI: 10.1163/157006010X489793
As a physically impossible event – after all it can be 'a miracle' or a prophecy or an allegory – this leaves still open the literary and religious tradition to follow for the alleged royal Indian witness. As such Muslim sources claim:
The literal splitting of the moon is a miracle attributed to Prophet Muhammad by early traditions transmitted on the authority of the companions of Muhammad such as Ibn Abbas, Anas bin Malik, Abdullah bin Masud and others.
Many Muslims, to counter the fact that he had apparently performed this miracle without any of the great astronomy/astrology cultures like the Chinese, Indians, Persians, Romans and Greeks noticing, have come up with the legend of Cheraman Perumal, the last King of Malabar (now Kerala).
According to Muslim legend, Cheraman Perumal (aka Chakrawati Farmas) was supposed to have witnessed this splitting of the moon.
WITNESS OF MOON SPLITTING (A MIRACLE OF PROPHET MUHAMMAD (PBUH)
CHAKRAWATI FARMAS KING OF MALABAR, INDIA
The incident relating to King Chakrawati Farmas is documented in an old manuscript in the India Office Library, London, which has reference number: Arabic, 2807, 152-173. It was quoted in the book "Muhammad Rasulullah," by M. Hamidullah:
"There is a very old tradition in Malabar, South-West Coast of India, that Chakrawati Farmas, one of their kings, had observed the splitting of the moon, the celebrated miracle of the Holy Prophet (pbuh) at Mecca, and learning on inquiry that there was a prediction of the coming of a Messanger of God from Arabia (Detail given below), he appointed his son as regent and set out to meet him. He embraced Islam at the hand of the Prophet, and when returning home, at the direction of the Prophet, died at the port of Zafar, Yemen, where the tomb of the "Indian king" was piously visited for many centuries."
The old manuscript in the 'India Office Library' contains several other details about King Chakrawati Farmas and his travel.
Others retell slightly different angles to the story.
It is said that
The Indian manuscript is kept in the India Office Library, London, which has reference number: Arabic, 2807, 152-173. It is quoted in the book “Muhammad Rasulullah,” by M. Hamidullah(2)
It is indeed catalogued there as a manuscript:
IV Foll[ios] 81-104. [Qissat Shakruti Firmad]. A fabulous account of the first settlement of the Muḥammadans in Malabar, under King Shakrûti of کوڈنگلور (Cranganore [Kodungallur]), a contemporary of Muḥammad, who was converted to Islam by the miracle of the division of the moon.'
For a different take and more detail on the veracity of the alleged event, SkepticsSE has a related question.
Dating the anonymous manuscript and interpreting its significance and meaning is quite difficult, but it was tried at least
In the sources quoted by historians of South-West India recurs time and again the personality of a Hindu ruler of Kerala, named Cheruman Perumal, who became convinced of the truth of Islam, divided his country among several rulers, departed for Arabia with the intent of performing the hajj and died on his return journey. The date of these events is a matter of controversy.
Some historians, following mainly the 16th century Arab writer Zayn al-Oin ai-Ma'bari, think that the events referred to above took place in the beginning of the 9th century A.D. However, many objections have been raised against this opinion and one of the historians claims that the conversion of the king could not have taken place before the 15th century. According to still another opinion, the conversion of the ruler was not to Islam but to Budhism and it took place between the fourth and the sixth century.
The present writer, not being conversant with the Malayalam and Tamil sources on which historians of Kerala base their opinions on the matter, would not like to venture an opinion in this controversy. The nature of the sources available at present seems to render the probability of a satisfactory solution to the purely historical problem very remote in any case. The intention of this article is to present a tradition which is almost unanimously rejected by modern historians, but seems to be popular among the Muslims of Malabar and may shed some light on the way in which they view their early history.
It is likely that the tradition came into existence in order to prove the antiquity and respectable origin of the Mapilla community. By depicting the first convert from Malabar as a king who was honoured and converted by the Prophet himself, at a very early stage of his mission (before the hijra!), was his companion for five years and died as a Muslim on his way to spread Islam in his homeland, the tradition was likely to enhance the status of the Mapilla community vis-a-vis other Muslim groups.
–– Yohanan Friedmann: "Qissat Shakarwati Farmad: a tradition concerning the introduction of Islam to Malabar", Israel Oriental Studies, Vol 5, p233–258, 1975. (Contains an Arabic transcript of the 'Qissat' with critical apparatus:)