AFter Bela Kun's communist regime was overthrown Horthy and his regime represented a legitimist reaction to the republican and socialist excesses that immediately followed the colapse of the K und K regime. Declaring a monarchy was a means of giving the reactionary regime "legitimacy", but the allies would never allow a Habsburg restoration. Karl made two attempts to gain the throne in Hungary and was booted out each time. There was no real alternative candidate so the fiction of a Kingdom without a King was the best solution for the aristocrats in power. There was some noise in the mid 1920's booming a British press magnate, Lord Rothermere, as a potential King of Hungary. Rothermere was a supporter of Hungarian irredentist claims, which was the one issue that could unite all sectuions of the Hungarian political classes. Other than that I don't recall any interest in filling the throne since the regency fitted the needs of Horthy and his associates the same way declaring Spain a monarchy with a vacant throne suited Franco.
Given that Hungary was the most agressively revanchist of the successor states and was governed by a reactionary, aristocratic clique, it is hard to imagine any scenario where Hungary does not voluntarily join the Axis, especially once Czechoslovakia was on the carving block.