One of major problems which Kingdom of Hungary historically had were its bickering nobles, who were crucial for providing military service, yet oftentimes refused to do so and also constantly bickered among themselves.
How much better would the Kingdom have done if it had - through contact with Roman Empire or in some other way - instead copied Byzantine themes as sociopolitical-sociomilitary model of organization? Specifically in cases of Mongol and later Ottoman invasions?
With the first Mongolian invasion of 1241 it would not make noticeable difference because King Bela raised a sizable army of a reasonably high quality and that army operated reasonably well in the field until it was outmaneuvered and outperformed. An alternative military system would not provide Hungary with a general of Subotai’s level and would not prepare its troops to fight against Mongolian style of a warfare. No matter how raised, it would still be lacking equivalent discipline, experience, easiness and speed of maneuver, etc. It seems that atbtyat time the Hungarian troops still had been relatively “light”.
As far as the second invasion of 1285 led by Nogai and Talabuga is involved, this was a badly organized looting expedition: Nogai was a far cry from Subotai in the strategic planning and he completely failed to synchronize operations of his and Talabuga’s forces being concerned only with his own looting. The main winning factor for the Hungarians were the numerous
stone fortified places built after the 1st invasion. With the Mongolian forces being smaller than in 1241, of a lower quality (a big part of them had been Kipchaks and the invaders had been noticeably short on the siege “artillery”) and having a different goal (this was seemingly a purely looting expedition) these fortifications became a serious factor: during the Western Campaign the Mongols had been simply bypassing most of the resisting fortresses. To make the long story short, the Hungarians did just fine with what they had at that time: reforms conducted after the 1st invasion were quite serious. To quote from Wiki, “In 1248, he declared the country's middle strata could enter a baron's service, on the condition that the barons lead the men on his land properly equipped (in armor) into the king's army. Documents from the time state that "the nobles of our country can enter into military service of bishops in the same way in which they can serve other nobles". After 1250, free owners of small or middle sized estates serving directly under the king were included (along with barons) in the nobility. Finally, new settlers were given "conditional" nobility in exchange for the requirement of fighting mounted and armored at the king's request.”
Now, as far as the Ottomans are involved, it seems that King Matthias found an answer in creation of a standing professional army well equipped with the firearms. Unfortunately, he did not use it to subdue the Hungarian magnates (and thus solving a problem of its financing), being engaged in the endless wars with the neighbors. By that time the merits of the Byzantine model became quite questionable against the Ottoman-style army with a core of a professional well trained infantry with the firearms. Needless to say that the still powerful magnates did everything to dismantle the Black Army during the reign of Matthias’ successors. Quite similar to what was
repeatedly happening in the PLC: Batory, Wladislaw and Sobieski had been conducting the short-lived military reforms pretty much killed by Sejms. It seems that as a prerequisite you’d need a strong hereditary royal power
domnating the nobility instead of being financially dependent from it. Specifically school of a military system would depend upon the circumstances.