Is that something that can be replicated in Hungary and Slavic areas in Austria for the purposes of this WI?
Its a complex subject as mentioned. Every society is different, and has different reasons for why it undergoes a demographic transition, even if there is a general correlation (although as mentioned, the causation aspect goes through several chains before arriving at effect) between increasing prosperity and wealth and then the fertility transition. At what level this starts for a society though, can vary a lot. France was the first country to undergo a fertility transition. Other countries didn't undergo a fertility transition for much longer, despite later being higher than the level of economic development that led to the transition in France. It might have been too in France, a consequence of the immense casualties from the Napoleonic Wars, which attenuated what demographic growth did exist via the demographic imbalance between men and women (it takes up to a century for major demographic events to fully go away - thus why even countries with negative TFRs will continue to grow, due to still having a bulge of youth left over their demographic booms).
So for why France underwent its demographic transition so early, it seems like, and I'm on more unsteady ground here, but I think it is what is backed up by demographic research - is an early secularization and de-christianisation. I had recently finished reading
The State in Early Modern France which had provided quantitative evidence for the decreasing importance of the Church in French life, and of its declining political authority, with various theological debates that gradually undermined its position. In general, it seems that the forces of order, the established, traditionalist classes, were discredited, and this enabled the spread and the adoption of fertility control. The Church didn't vanish from French life of course, and it later recovered much authority, but it never seemed able to put the fertility control genie back in the bottle. Perhaps economics had something to do with it, but it was this ideological war which the cultural establishment constraining it lost which resulted in the relatively sudden switch in the French position - this economics land reform was also something which came from these elites being discredited too.
Now to try to apply this in Hungary and Slavic regions, on the French model, I'd suggest that there would need to be the same thing done. Traditional religious, "feudal", and hierarchical structures need to be discredited, to enable the transmission of the idea and the practice of fertility control throughout the population. This seems to me to be quite difficult in the conservative and traditional Hapsburg empire, and I don't know what sort of system could lead to the Church undermined to the same extent to enable that transition. In France, secularization was a complicated and lengthy process, which was particular to the French situation - relying upon the conflict between the French church and the peope, the debate over Jansenism and Gallicanisme and the papal bull Unigenitus Dei Filius which raised an incredible hellstorm of literal decades of intense political debate over it and which set forth the effects which destroyed the monarchy. Having any such event occur in the Slavic regions of Austria-Hungary and the Slavic regions alone is something that I can't see happening in light of the way Hapsburg administration was set up : indeed, how to have the event even happen at all would be difficult. There's a reason why France was quite unique.