IOTL, the Habsburgs fought - and were defeated by - Revolutionary\Napoleonic France in no fewer than 4 different wars between 1792-1809 (First, Second, Third, and Fifth Coalitions). Surely the French could see a trend there. And indeed it would join the Sixth Coalition once France became weakened enough. The French were able to march deep into Austrian territory in several of those wars, and even occupied Vienna twice. So I suppose it's reasonable to say that the Habsburgs were defeated quite decisively and even were at several points at the mercy of the French. Following the Fifth Coalition specifically, the Austrian Empire was punished quite severely, losing a lot of territory and forced to become an ally of France. Since Napoleon could probably see that the Austrians may at some point join another coalition if they saw an opportunity, could he have decided instead that it's preferable to end the Habsburg Monarchy for good, breaking it up into the several countries that make it up? There are a few issues here with this:
1. Are the Habsburgs truly so thoroughly defeated in late 1809 that the French could simply enforce whatever peace they want, or could they still fight on? They still have troops on the field, but much reduced and with a lot of territory taken - including Vienna itself. Their allies can't really help them, but there's Russia to the east whose not at war with France yet. But Russia is at this point at war with Britain and the Ottomans, and has only just wrapped up its war with Sweden, so they possibly might not be relied upon in this situation.
2. What are the international repercussions for such a move? Obviously most of Europe would be enraged by this, but are the repercussions really severe enough to dissuade Napoleon from breaking up Austria? Most of Europe at this point is either already at war with Napoleon or allied to him (notable exception being Russia), so it's not like a new coalition could form to deter him (Russia could join in, but as mentioned this has its own problems for the Russians), but on the other hand Napoleon needs the trust of the rest of Europe if he wants to remain hegemonic.
3. Say Napoleon does decide to break it up, how does that work? Presumably all of Galicia goes to the Duchy of Warsaw, and Hungary is made independent. What about Bohemia/the Czech lands? What about Croatia, does it remain part of Hungary or is it made into a new "Kingdom of Illyria" or something? Which o Napoleon's relatives is given which, or perhaps some local noble, or some German prince are granted one of those territories? Finally, what would be done with the rump of Austria proper, still under Habsburg rule or someone else?
1. Are the Habsburgs truly so thoroughly defeated in late 1809 that the French could simply enforce whatever peace they want, or could they still fight on? They still have troops on the field, but much reduced and with a lot of territory taken - including Vienna itself. Their allies can't really help them, but there's Russia to the east whose not at war with France yet. But Russia is at this point at war with Britain and the Ottomans, and has only just wrapped up its war with Sweden, so they possibly might not be relied upon in this situation.
2. What are the international repercussions for such a move? Obviously most of Europe would be enraged by this, but are the repercussions really severe enough to dissuade Napoleon from breaking up Austria? Most of Europe at this point is either already at war with Napoleon or allied to him (notable exception being Russia), so it's not like a new coalition could form to deter him (Russia could join in, but as mentioned this has its own problems for the Russians), but on the other hand Napoleon needs the trust of the rest of Europe if he wants to remain hegemonic.
3. Say Napoleon does decide to break it up, how does that work? Presumably all of Galicia goes to the Duchy of Warsaw, and Hungary is made independent. What about Bohemia/the Czech lands? What about Croatia, does it remain part of Hungary or is it made into a new "Kingdom of Illyria" or something? Which o Napoleon's relatives is given which, or perhaps some local noble, or some German prince are granted one of those territories? Finally, what would be done with the rump of Austria proper, still under Habsburg rule or someone else?