Consider the medrash that Moshe Rabbeinu was the king of Ethiopia. If you want to say that since it was before Sinai, it would not apply, consider the Khazars where the king and nobility appear to have converted to Judaism. The child of Esther and Achashveiros was halachically Jewish.
Esther later life
She is recorded as being the Queen of Persia for years to come and the
mother of the next Persian King Daryavesh, who eventually allowed the
Jewish people to return to Israel and begin rebuilding the Temple and
their independent lives in the Holy Land.
Munbaz was also a Jewish convert who ruled a kingdom.
King Munbaz of Adiabene, son of Queen Helena, was a convert to Judaism
(and not king of Israel). The Talmud in Baba Basra (11a) tells how he
gave all of his personal wealth to charity.
There is also the legend that Rabbi Shaul Wahl (c. 1542-1622) was appointed king for one day while the Polish council deliberated over which of the contenders to the throne should be appointed. The legend is that he was the only person they trusted to abdicate once a decision was made. Even if it is not true, the fact that there is such a legend means that it was halachically acceptable.
Wikipedia on Yemenite Jews says
There are also several historical works which suggest that a Jewish
kingdom existed in what is now Yemen during pre-Islamic late
27 "The story of the Jews, finding the words" by Simon Schama. part
two, chapter 6 "Among the believers" page 233 "By the late fourth
century CE, just as life for Jews in Christendom was beginning to turn
starkly harsher, Judaism made its spectacular conquest in Arabia, when
the kingdom of Himyar (corresponding, territorially, to present-day
Yemen, and the dominant power on the Arabian peninsula for 250 years)
converted to Judaism. For a long time, it was assumed that the Himyar
conversion was confined to a small circle close to the king- Tiban
As'ad Abu Karib, the last of the Tubban line, - and perhaps included
the warrior aristocracy. There is still a lively debate regarding the
extent of Himyar Judaism; but the evidence of both inscriptions and,
more significantly, excavations at the mountain of the capital of
Zafar, which have uncovered what seems likely to be an ancient mikveh,
suggests to many recent scholars (though not all) that the dramatic
conversion was more profound, widespread and enduring.