Hungarians going the Bulgarian way is one possible solution
How about Hungarians being Germanized? Let's say Christianity doesn't catch on very well or that Stephen I does not manage to defeat the pagan rebellions alone and is forced to rely on (even heavier than OTL) German support. Something akin to the Northern Crusades happen and once the pagans are defeated Hungary is not only left with a lot of German knights becoming nobles, they also invite lot of German settlers into the basin. Over the next couple hundred years this becomes a trend and Hungary becomes more and more like OTL Prussia, but way earlier, and with a bigger head start: lots of arable land, lots of resource rich mountains that are a good incentive for heavy German settlement. Mongol invasion also spares most of the German population (living in fortified cities) and goes heavy on the rural population, which is again supplanted by even more settlers from the HRE. Maybe the whole kingdom becomes a subject to the HRE, if not full member.