Then just set yourself up as the feudal overlord of western Europe, convert to Catholism etc, make yourself at the top of the existing political system. Have your base of operation in Hungarian plains and a capital in Budapest or somewhere deeper within the European heartland like Aachan, or even Rome (!!) while the rest of western Europe swears fealty to the Le Khanate.
While I wholeheartedly agree with most of your points I would like to disagree on some issues.
As I previously said the most obvious POD to make the Mongol conquest of Europe possible is to give Ogedei Khan the health of Winston Churchill to hold his liquor and live to the old age. Say 10 years longer than in OTL.
There are at least two scenarios after Europe is conquered by the Mongols.
Scenario #1 - Batu is undisputable head of Jochi Ulus including Europe. Actually this scenario is not that obvious as it may seem which I'll explain later. In this case I seriously doubt that capital of the Golden Horde might be in Aachan or in Rome. Look at the map of the Jochi Ulus - Europe is the extreme Western corner of this Ulus. And Mongol Europe will be much poorer and far less populated region than it was in OTL.
The capital would be wiser placed somewhere between Eastern Hungarian plains and Volga steppes. As a variant - two capitals, one in Hungary and the other on Volga; half year the Khan spends in one capital, the other half in the other, depends on which side of the ulus needs his attention more at the moment.
We must understand that steppes will be the core of the Golden Horde, its power base will be there, nomad mounted archers, backbone of the military might.
As for the Jochi Mongols becoming Catholics... Well, having more Christians in the entity makes it more probable, no doubt. But I am of the opinion that the steppe population of the Golden Horde was already inclined to Islamization and that solved the matter in OTL and would play the same decisive role in ATL.
But if the Khan decides to get Christian - why Catholicism (or to be more exact "Western Christianity"), why Rome?
As in this scenario the Balkans will be conquered and Constantinople will be freed from the "Latin occupation". So the Khan might become a Christian more inclined to the "Greek, Orthodox" Christianit rites. Well, the Khan might take a role of Constantine and unite all the Christians together, so there will be no Western, Latin or Eastern, Greek Christianity.
The Mongols could be quite persuasive, you know.
As for the Khan becoming the top of the existing system. With the political centre of the Horde being in the steppes from Hungary to Central Asia the main legitimization of the Mongol power is "Chengizz Khan gave the Jochi House these lands and that's why we own them. Point." Nothing more.
Europe being in the corner of the ulus as the source of income, loot and slaves will not be that important to care what they think about what is legitimate and what is not.
You know in OTL the conquered Russians called the Khan - "Tsar" (derived from "Caesar") though the Mongols did not give a damn what the Russians called him. But that was the highest known title the Russians knew. I guess the conquered Europeans in this ATL would call the Mongol Khan "Emperor", Kaiser, Caesar or something like that without any coronation or anything. Just out of respect and fear.
Scenario #2.
Ogedei had good relations with his nephew, Batu, who was in charge of the European invasion and the conquest (together with Subudai). But the numerous princes of other Houses including his own sons would find a way to explain to the Great Universal Khaan that Batu took too much power and influence into hands after the conquest would be over. Actually Batu will be the most powerfull man in the Empire after Oghedei himself. That must not be tolerated.
And there was a way to solve this problem. You know Chingizz Khan gave these lands to the Jochi House, that must be that way, that was the highest authority, no way to change it. But this ulus might be divided into several parts (hordes) headed by other sons of Jochi and thus more dependent on the Great Khaan Ogedei as arbiter.
Something like that happened in OTL. The eldest son of Jochi, Ordu, the elder brother of Batu, was given his Horde in the Eastern corner of the Jochi Ulus, White or Blue Horde if I remember correctly. And the linage of Hordu headed this shithole of steppe for more than a century autonomously. We do not know, that might be the initial intention of Ogedei - to divide Jochi Ulus among Jochi sons. The concentration of power was essential during the conquest, but it was dangerous for the central imperial power afterwards.
So Ordu in this ATL might be given Europe as his Horde. Why not? He was not as talented as his little brother, Batu. But he was definitely not a fool as a general and a politician. So this Horde would definitely include Hungary as the best place to settle Mongol migrants from Mongolia proper and some other loyal nomad tribes ofdifferent, mostly Turkic origin. That will be power base of the Horde. To get some chunks of the Western Cuman steppes would be fine.
It would also be advisable for the Great Khan Ogedei to put some famous and influential general from the old guard of his father, Chengizz Khan in place -to have some direct imperial influence.
Subudai would be the best choice - Ordu was the elderst son of Jochi, the eldest son of Chengizz Khan, that would be quite honorable for the old man to serve him. This best general of the era would cement the conquest and keep an eye for the Great Khan decision to divide Ulus among brothers. But later when the things settle down Subudei will be needed on other fronts of the World Mongol Empire.
So this European Horde will be what you may call Le Khanate
Here the Mongols would need to put themselves into the existing chain of legitimacy, becoming Catholics, crowned by the Popes, Khans styling themselves as Holy Roman Emperors, etc.
