One factor that could play a role in shaping a different South Slavic unification, if we were to look at the Illyrian movement, is to somehow deal with the Illyrian language conceptions. My familiarity with it, and in fact the Illyrian movement in general, is rather limited, so I'm just going on based off of what Wikipedia says in English, Serbian, Croatian, and Serbo-Croatian.
As long as the Illyrian language conceptions end up moving from Kajkavian and Čakavian to Štokavian, and in turn going from etymological/morphonological to phonological orthography (aka going Vukavian/Vukovian), things are going to lean towards the Serbs being the major agitator for unification, whether it be in the form of a Greater Serbia or in the form of Yugoslavism, especially if we see the murder of Hacı Mustafa Şinikoğlu Paşa in late 1801 as per OTL, leading to the tyranny of the renegade Dahije and the subsequent First Serbian Uprising, as at that point the ball is just going to start rolling.
Preventing the move towards Štokavian is hard to do, however - while the Serb intellectuals in the Habsburg Empire didn't seem entirely interested in the developments with Illyrianism, Štokavian by this point had become the widest spread narečje (bigger than a dialect, so I'll use this for clarity) within the region of the western Balkans, and anyone seeking the cultural and linguistic unification of the South Slavs would want to head towards using it instead, which is what Ljudevit Gaj ended up doing, even going so far as to try and make it more accessible to those speaking Kajkavian and Čakavian. Similarly, preventing the shift to phonological orthography is hard - even if you were to have Vuk Stefanović Karadžić die prior to 1810, after growing ill and being unable to obtain medical treatment, from what I can tell, the choice of etymological/morphonological orthography was panned by some contemporaries, even if the idea of it lasted the longest in comparison to all other Illyrian conceptions. And this isn't even considering the issue of Hungarian resistance to the Illyrian movement.
The only way you could turn things around, at least that I can think of at this moment, is to have the Hungarians use the movement as a means of dividing the South Slavs rather than unifying them, stepping behind the aristocracy in going against both Gaj's shifting away from the use of Kajkavian as a literary language and Karadžić's own efforts, creating a divisionary counter-culture. From there, on the Illyrian end, have Kajkavian and Čakavian (at the time both linked to some degree with Croatian and Slovenian) undergo their own sort of Vienna Literary Agreement, leading to the creation of the etymological/morphonological Illyrian language. Meanwhile, on the Serbian end, aligning with the Austrian Serb intellectual circle and general upper class, and the Serbian Orthodox Church, begin the standardization of Slaveno-Serbian.
Now, of course, the question at this point is how does this all relate to a different South Slavic unification, if the Croats and Slovenes have been divided from the Serbs (not discussing anything that could happen in regards to Bosnia, which may even lead to some Ikavian-using, Arebica-implementing Bosniak)? Well, even with all this in mind, and a stronger (albeit forced upon) Illyrian identity, Yugoslavism was bound to rise one way or another. The only difference here now is that the Hungarians have been weakened in exchange to some of this being contained, though that could all be countered if we have the Ausgleich occur as it did OTL. You'd either need to avoid it happening, or have Illyria be raised at the same time as Hungary (the latter perhaps receiving more sovereignty/autonomy in exchange for the loss of territory) in a form of early trialism. From this point onwards, you'd just need to maintain Serbia in the Habsburg sphere, which will require some concessions on the Habsburg end as to prevent mass support in Serbia for aligning with Russia (as well as any controversies that could be used as an impetus for a coup, ala Alexander Obrenović).
With Illyria under the Habsburg monarchy, and the Obrenović Kingdom of Serbia in the Habsburg sphere (and possibly even a member of the Triple Alliance, now the Quadruple Alliance), in the event that we do see a Great War and the Central Powers lose again, we'd have the fracturing of the Habsburg Empire, a bruised Serbia (Bulgaria likely being part of the Entente this time around) and a demolished Montenegro (as it likely would've gone Entente too, and have been beaten as easily as OTL), an independent Illyria (which has lost some of its land to a side-switching Italy) is in the prime position to propose the incorporation of the Kingdom of Serbia and force the incorporation of Montenegro (ala the Podgorica Assembly), and establish a larger Illyria, hellbent on getting back the territories it had lost from Italy, as well as those lost by Serbia to Bulgaria (and likely looking at Romania too if the entire region of Banat fell under its control, including the Slavic parts owned today by Serbia, assuming of course we didn't see an equivalent to the People's Administration for Banat, Bačka and Baranja, which a newly independent Illyria could take advantage of).
