Well, IOTL, he named his son Istvan the “Deputy Regent” before he died in a mysterious plane crash, which indicated he had some sort of succession plan to make the Regency a hereditary office,
Ok. That would be possible thing. In other hand butterflies might change Horthy's plans. Since he died in 1957 he could make something different like appoint Otto von Habsburg as his successor. Of course him should ensure that France and Britain and speciality neighbors are fine with that. But since OTL Entente was pretty willingful to accept Hitler's actions to reduce Treaty of Versailles they hardly care if Habsburgs return to Hungarian throne as long as it doesn't mean return of Habsburg Empire.