That gives me an idea: maybe political Catholicism would have become the glue that holds the empire together. The Hapsburgs would point out that ethnonationalism is barbaric and pagan, and a multinational, multiethnic empire where everyone (Catholic or not) acknowledges Catholic social teaching is best at fostering peace and unity. Maybe the empire declares Latin the language of the bureaucracy to further strengthen this legitimacy.
That would win over the loyalty of the Poles, Galicians, Slovaks, and Croatians.
German-speaking Austrians would be encouraged to see themselves as Catholics who speak German, and who enjoy the freedom of worship in contrast to Bavarians who suffer through kulturkampf.
Romanians would be encouraged (through various subtle measures) to join Romanian Greek Catholic Churches to create a distinction with the people on the other side of the Transylvanian mountains. Over time the Transylvanians would see themselves as separate from the "other" Romanians, the way that the Flemish see themselves as separate from the Dutch.
Hungarians would be a "nation within the nation" a similar way that Quebec is within Canada. The average Hungarian would in general only interact with institutions bearing the crown of St. Stephen, so this ideology would be toned down within the Kingdom of Hungary.
There would be subtle social pressure for non-Catholics to convert to Catholicism if they wanted to seek higher political office.
I do see scope for conflict with this ideology, though. First, the Russian and German empires will constantly suspect AH has designs on Poland and southern Germany, respectively. Second, AH would be seen as an ideological threat to the anti-clerical regimes of France and Italy. Third, while the Poles will prefer living under the Hapsburgs than the other two empires, they will still prefer that *all* of Catholic Poland be united - either under the Hapsburgs or independently. AH will need a powerful ally to guarantee its security - and what better one than the UK, which absolutely loves such games?
Maybe the Russian Empire will intentionally encourage Jewish emigration to AH, since they now have a convenient "dumping ground" for their "problem". Maybe they would even think this "problem" would undermine an imperial rival.
If that takes hold, then my previously described ideology would come under pressure. A refugee influx of that scale would cause tension in even the most tolerant and liberal societies.
Not going to work. At least not the way you propose. First there are a lot of othodox people in the Empire, not just the romanians - like serbs and part of the ukrainians. Than there are the jews and the muslims of Bosnia, and there are lutherans and calvinists in Hungary. Pushing a catholic agenda wont go down well with any of these.
Whats more we are far too late in time for religion to be the main identity forming element for the masses in Austria. The alternate for nationalism by this time was socialism which wanted people to identify along class lines instead of nations - and it failed completly by WWI. Maybe if we got an earlier and stronger christian socialist movement (it was already strong in Austria) that could have a positive effect - but I dont think that you can supplant nationalism as one of the defining ideologies - you would need a much earlier POD.