I think the Kingdom of Hungary adopting Orthodoxy wholesale is a bit too late. You need to make a change rather early, due to the fact that the early Hungarian state built on the existing church infrastructure that was already in place when they arrived. Church infrastructure that was inherited from the recent two, two and half centuries before the founding of the kingdom.
As already noted, the schism in the church happens in 1054, and the kingdom is established for half a century by that point (from 1000 AD onwards). Those developments sort of pass each other by, further complicating matters. I think you need to go before the existence of the kingdom, back to the OTL roots of church infrastructure and adoption of Christianity in central Europe as a whole, to really get a good idea on how to build a predominantly-Orthodox ATL outcome.
The Kingdom of Hungary inherited a substantial part of its future churchly infrastructure from the days of Great Moravia. Though there is a century between the small empire (a principality) and the kingdom, the various churches, clerical staff and monks survived the disintegration of the old empire. Previously, when the empire adopted Christianity officially in the early 9th century (830s), there was a bit of a tug-of-war between missionaries and church officials of the Latin tradition and those of the future Orthodox tradition.
There were handfuls of Frankish missionaries in GM before, representing the Latin tradition, while Cyril and Methodius were Byzantine missionaries who represented the eastern tradition, the future Orthodox. There was still cooperation between them and Rome, with the pope no less, but it is hard to deny the ongoing tension in GM between the two traditions. The empire was sort of pivoting both ways, but when Christianity was adopted, it was leaning more pro-Byzantine at that time (while still wanting good relations with the Franks). The prince that followed
the one who invited Byzantine missionaries, brought somewhat of an end to that tension.
He reoriented the country in a more pro-Latin, and consequently more pro-Rome and pro-Frankish direction, while still doing independent politics and some territorial expansion of the realm.
GM bit the dust by the end of the 9th century, but local warlords and noblemen preserved the church infrastructure within the scope of their fiefdoms and kept it going. These successor noblemen (e.g. Hunt and Poznan) later merged with the beginnings of the Hungarian state, as vassals of its princely and later royal family. The Hungarian state inherited the church infrastructure via these guys. Churches and even a few monasteries didn't vanish with the empire, they kept chugging on. No doubt the officials in Rome or those in the bishoprics of neighbouring early German states had an interest for Christianity to continue in the whole area, business as usual.
Finally, in today's western Hungary, there was substantial Slavic settlement in the early medieval period, even though the people there never formed a bigger polity of their own. They later became allies of the east Franks and some were a bit closer to them, as they didn't have the same degree of political and military independence as the Slavs in the Great Moravian monarchy to the north. These areas in western Hungary had adopted Christianity too, but they and their church infrastructure always seemed to be pro-Latin (and pro-Frankish) in orientation. One of the more powerful local noblemen was
prince Pribina, exiled from GM during the formation of that state. (His original principality was annexed and his line was overthrown.) He was already a Christian at the time of the move to his southward exile, and while still on the throne, he promoted a pro-Latin course for the beginnings of the church in what would become GM. While in exile in the western Hungarian areas, him and his heirs (Koceľ, etc.) built a fair amount of new secular and church infrastructure in the region, during the course of the 9th and 10th centuries. Eventually, said region was incorporated into the fledgling kingdom of Hungary.
As you can see, there is a definite orientation towards the Latin tradition of the Church (what would eventually become Roman Catholicism by the later Middle Ages) in the early history of Hungary. This is mainly because most local supporters of the church and the creation of church infrastructure leaned towards Rome in their religious and political orientation, even early on in the history of the adoption of Christianity in central Europe. This was a trend throughout the whole 9th and 10th century. The pro-"Orthodox" leaning in central Europe by a West Slavic ruler was a rather short-lived experiment. It only lasted during his reign, for a few years, with a pair of renowned missionaries. However, even that wasn't done out of some fondness for the Byzantine way of doing things. It was done out of political expediency. That particular prince, while getting on with the fdranks, wanted tobuild stronger ties with the Byzantine emperor. So, as a gesture of diplomatic good will, he asked him to send him some missionaries from the Byzantine Empire. The emperor accordingly sent his two men from Thessaloniki (Salonica), who also started developing a writing script (an ancestor of later Cyrillic), on order to ensure the Byzantine Empire would have a cultural influence on the would-be West Slavic ally neighbouring the Franks. Ultimately, neither the Byzantine church influence, or the script influence provided by the missionary duo, would last. The very next GM prince alone would ensure a more pro-Latin/pro-Frankish direction.
And this brings me to Hungary. The best way to ensure a predominantly Orthodox or all-Orthodox Hungary, is to have all of the adoption of Christianity and church infrastructure in early medieval central Europe take place under the aegis of a pro-Byzantine direction. Easier said than done, given the proximity of the East Franks, and their political and economic influence. Franks and Byzantines were the two European "superpowers" of the time, as the main heirs of Roman antiquity, and their influence on smaller polities developed accordingly. The Franks are next door to the West Slavs and their early states and the territory of future Hungary. The Byzantines are, comparatively, quite far away. Even with central European rulers adopting the Byzantine flavour of Christianity and completely excising the Latin rite and tradition (
somehow), the fact of the matter is that buddying up with the Byzantines is prohibitive in terms of logistics.
Ironically, in OTL, Hungary under the Árpád dynasty still sought good relations with the Byzantines (even some royal marriages), but that was for different reasons. Orthodox-esque cultural elements of Christianity didn't vanish in the early centuries of the Hungarian state. But they were always a minor thing and partly a result of some past heritage, or appeared because of the influence of eastern neighbours who
had adopted Orthodoxy. Turning towards the Byzantines for more than political/diplomatic reasons, turning towards them for reasons of religious reorientation of the state, would be difficult in later centuries. And I do mean difficult. If you have, say, two centuries of building the domestic church on the basis of a Latin orientation and an orientation towards Rome, it isn't plausible to just throw all of that out and make Hungary Orthodox, just because some ruler of the Árpád dynasty decides so. A later Orthodox reorientation gets even more difficult in later centuries, and especially after the extinction of the Árpád dynasty. 14th and 15th century Hungarian dynasties didn't really care that much for political ties with Rhomaion anymore, especially in light of its gradual diminishment at the hands of the crusaders in the Fourth Crusade and then the Ottoman Turks.
Also, have a look at
this older thread, it discussed the same topic.