Would Hungary then be governed similar to how the Southern (ex-Confederate) US was governed during Reconstruction, with the Hungarian minorities playing the role that blacks, scalawags, and carpetbaggers played in the Reconstruction-era Southern US?
I very much doubt that would be necessary. For one, unlike the Southern states, which were (with the exception of West Virginia) kept in the same territorial form, the dualist stucture of the Empire would be wholly dissolved, and the non-Hungarian regions of the former Hungarian half would all be immediatised.
"Hungary" would thus be a territorially reduced member state of the Empire, with no authority over Slovakia, Transylvania, Ruthenia or Croatia. (I assume that Vienna wouldn't want the Romanians
too powerful, so I think Hungary would be allowed to retain the Székely Land, and probably a land strip stretching towards it.)
Furthermore, the Hungarian elites most interested in maintaining the old status quo would be replaced by more co-operative reformists, who'd get to carry out their 'domestic' agenda for Hungarian politics in echange for accepting the new reality.
The resulting Austrian Empire would either be quasi-federal, or at least embrace a measure of political devolution. This would allow the Czechs, Slovaks, Ruthenians, Poles, Greek Catholic Ukrainanians, Hungarians, Romanians, Croats and Slovenes a degree of self-government over the internal affairs of their own constituent provinces/states within the Empire, and would almost certainly give all their languages official status (besides German, the 'imperial language') in all domestic matters.