Structure of an Austrian Germany versus a Prussian one

In 1848, the Hungarian Revolution is a success and a Hungarian Kingdom (Republic?) is established in the Pannonian Basin. Galicia and Bukovina become a separate Hapsburg Kingdom of Galicia and Croatia-Slavonia, Dalmatia, and Burgenland remain Austrian.

Austria, now focused almost entirely on Germany, pushes for German unification under Austrian domination. In the Austro-Prussian War of TTL, Austria defeats Prussia, annexes Silesia, allows for Prussian dismemberment (Hapsburg principalities established in Schleswig-Holstein, North Rhineland, and South Rhineland to compensate the dispossessed Hapsburg of Italy, and Saxony returning to Napoleonic borders), and is without a doubt top dog in the German Empire.

What would be the structure and culture of an Austrian Germany versus a Prussian one?
 
An Austria without Hungary is kinda doomed to be honest, why would they be in the position to defeat Prussia when they weren't IOTL with Hungary and its resources?

If Austria collapses in 1848 then the Erfurt Union has a stronger chance to pass through, getting you a very different Germany under Prussia in 1850-1851.

https://en.wikipedia.org/wiki/Erfurt_Union
 
If you want an Austrian Germany, you don't have to remove Hungary, you can either have a war break out at the Herbstkrise in 1850 or make the Austrian win in 1866, I prefer the former.
 
I'm not really the person to ask, but I think it's fair to assume that Catholicism will be a greater part of the 'national character' than Protestantism. I'm not sure exactly how that might affect the foreign policy of an Austrian-led Germany - religion had largely ceased to be of great significance in international relations by the mid-19th century. But the 'hard-working industrious Protestant' conception of Germans IOTL will obviously differ.

There would also be a greater continuity between the Holy Roman Empire and Austro-Germany, as each would be dominated by an Emperor generally seated within the Austrian Circle, particularly Vienna. This might generate a sense that Austria and Vienna have always been the 'natural' centre of gravity within 'Germany', perhaps influencing historical studies and biasing them towards the assumption that Austrian preeminence was inevitable. In turn this could lead to an over-emphasis on Austrian power in the Holy Roman Empire, perhaps leading to a view that the HRE was more centralised than it was in actuality.
 
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