In regards to NWAG's claim that embracing liberalism and nationalism would only doom the Empire, I think thats about fairly incorrect. I can't see why embracing liberalism inevitably dooms the Austrian Empire in the long run.
On the contrary, trying to push absolutist rule (especially under a German Emperor) could arguably push liberal nationalists towards independence as an easier method of escaping the rule of Imperial autocracy as an alternative to trying to reform a vast government system towards a democratic model against the wishes of a powerful and entrenched aristocracy.
Sure, you might try to push fundamentalist Catholicism as a unifying identity, but you've got to adapt that to the times and the NES - as SK said, political Catholicism is dead in this timeline, and trying to resurrect would be seen as desperate, sad, and ultimately pointless if the majority of your population thinks political Catholicism is a joke.
Ethnic separatist movements don't just manifest because "Emperor Joseph is not teh Hungarians." They are the result of an extended period of divergent policy desires between a dominant governing ethnic group (in the Austrian example, the Germans in Vienna) and a less dominant ethnic group (in our example, the Hungarians). If enough of the non-dominant group's policy desires aren't satisfied, then they will begin to wonder what they even get out of this union in the first place, and begin to consider and embrace ethnic separatism as a means to achieve their policy desires in the first place.
A liberal multiethnic federal union is an actual solution this problem, as it (theoretically, anyway, it really depends on who's running things and just how much of a true multiethnic federation it is) gives all ethnicities the opportunity to achieve at least some of their policy desires, likely compromising some of their policy ambitions for the political and military security and economic opportunities federation with other states brings.
The typical example of this is in education in your language - if you are able to set your own ethnicity's education policy, then you can educate in your own language (usually teaching the chosen 'national' language as a second language) then you are able to ensure that members of your ethnicity are able to receive a quality education and participate in all sectors of society. This is a very common complaint across history that leads to secession - a central government dominated by a single ethnicity forcing down a language in education and as a result members of non-governing ethnicities out of educational institutions because they don't speak the language of governing choice.
Multi-ethnic federations can be good for these types of problems, because they result in ethnicity being able to compromise and achieve some of their political ambitions. Case in point, India seemingly hasn't collapsed into a horrible civil war that leads to the dissolution of the union and the secession of Kerala, and its probably not going to barring a serious divergence in policy between New Delhi and the rest of the country. (This is not to say that violent popular separatist movements exist, but somehow the entirety of India hasn't completely seceded from the central government so yeah).
Obviously multi-ethnic federations (or federations in general) aren't perfect, and massive policy divergences can still appear and lead to secession (case in point, Yugoslavia). But they are
a solution and they have been shown to work. The key thing is that multiethnic federalism relies on the ethnicities being able to achieve the policy that they themselves want, which requires a degree of voting and thus, liberalism (because you can't really express what you really want from your government without free elections and political freedom). Thus, you need liberals writing your constitution: in an 1850s context, your government needs to embrace Liberalism.
Thats just one solution to the issue of a multiethnic state trying to stay together, I'm sure there are others. Doesn't matter, its still a method of keeping Austria-Hungary together while embracing liberalism as a governing ideology.
As an aside, in any case to keep a multi-ethnic union together you need to embrace nationalism anyway. You need to create a feeling of loyalty to the wider polity rather than your own ethnic group, because when people are more loyal to the desires of their ethnicity than the polity you start to get policy divergences. In any case, you need to build pan-Austrian nationalism, whichever route you choose in trying to keep your state together. Technically speaking, everything you do will be embracing nationalism.