Did a Jewish King of Narbonne exist?

It wasn't a kingdom. The idea is that there existed a Jewish principality in the Frankish realm, led by a Nāśī as vassals of the Carolingian kings. It is mostly advocated by Arthur Zuckerman, as you have noted. His theory has however been widely dismissed by historians in general.

Zuckerman's claim is that the Jews of Septimania aided Pepin the Short in his seven year siege to capture Narbonne from the Muslims. As a reward, the Jews of Narbonne was given extensive lands and privileges, and Makhir, a Babylonian rabbi supposedly from the Royal House of David was installed as their leader. Zuckerman argued that he created a dynasty of Jewish rulers related to the Carolingian kings by blood.

That a Jewish community headed by a Nāśī existed in Narbonne is not very controversial. Zuckerman's identification of those Jewish leaders with other historically recorded nobles of the region is however widely discredited. Specifically, Zuckerman argues that Makhir (the alleged founder of the Jewish dynasty) is the same person as Count Magnario of Narbonne and Aymeri de Narbonne. From that, he argues that Magnario/Aymeri/Theoduric/Makhir fathered Saint Guilhelm of Toulouse, founder the House of Guilhelmides.

The problem is that contemporary records don't support Zuckerman's arguments. In the historical record, Makhir is only mentioned in a 13th century gloss of Abraham ibn Daud's 1161 Book of Traditions. No other source names Makhir at all. His supposed dynasty's continuity from its supposed founding up to ibin Daud's time is totally unrecorded, and there is no way to verify its claim of Davidic lineages. Moreover, it is directly contradicted by the contemporary (with the gloss) Gesta Karoli Magni ad Carcassonam et Narbonam. The Latin text states that the Jews of Narbonne had, when the city was captured from the Muslims in 759, an existing leader of the Davidic line. As opposed to one imported from Babylon by the Franks.

There doesn't seem to be any actual evidence behind Zuckerman's claim that Aymeri was the same as Magnario and Makhir either. His primary evidence is that a certain Magnario was mentioned in Narbonne's 791 charter, the original of which having been long lost. Based on fragmentary facsimiles of engravings that survive in the 1681 De Re Diplomatica, Zuckerman claims the Magnario it mentioned is actually Maghario. According to him, the latter is Makhir in Latin.

That is, he argues the (as most people read it) gn in the word is actually gh. There is however a partial word regna from the same document. Other historians generally took the visual similarity between the two gn's to mean that the one in Magnario is, well, also gn after all. In this they are backed up by Magnario being the Latin of a known Germanic name. The same name, actually, as the significantly better attested Meginarius, notary of Louis the Pious.

Lastly, a note about his claim that Saint William of Toulouse was Jewish. Zuckermans cites as "evidence" Ermold the Black's Poeme sur Louis le Pieux. In that poem, Guilhelm of Toulouse did not fight on the Sabbath. Zuckerman claimed this proved he was Jewish. Most historians thought it was because his army was part Jewish

While Zuckerman's theories are interesting, they don't seem to be based on much in the way of facts. Sometimes the obvious explanations are more convincing than fantastical ideas that predicates upon references to this fabled Jewish dynasty being systematically purged from all historical records.

When you hear hoofbeats, think of horses, not zebras.

_{-- Professor Theodore E. Woodward}

Sources:

1. Cosman, Madeleine Pelner, and Linda Gale Jones. Handbook to life in the medieval world. Facts on File, 2008. Page 316.
2. Cohen, Jeremy. "The Nasi of Narbonne: A Problem in Medieval Historiography." AJS Review 2 (1977): 45-76.
3. Taylor, Nathaniel L. "Saint William, King David, and Makhir: A Controversial Medieval Descent." American Genealogist 72 (1997): 205-224.