Alternatively, if you want a later POD you could go for 1851 and the Hapsburg Winter. We tend to take it for granted the Austrians 'unified' Germany, but really it should be understood of them making the best of a bad situation after their dream of balancing hegemony in Germany and South-East Europe collapsed.
People do tend to forget, but the moment that the Austrians moved from the dominant leader 'of, but outside' Germany to being the leader of an Austria within Germany was incredibly dangerous. They tried to portray it as a great national destiny, but it could have fallen flat if they hadn't recognised they were on the verge of disaster and seized triumph instead.
At the- frankly, over mythologised- first Diet of the German People in Frankfurt, it was by no means certain that the first Emperor of Germany would be a Hapsburg. It also wasn't certain there would even be an emperor, but I won't go on one of my tangents about republicanism here. There were various other princes who might have taken the crown. The King of Bavaria had a real chance, as did the Saxons.
As it was, the collapse in Hungary and Venetia convinced Archduke Charles to make a virtue of necessity- but if the Hapsburgs had tried to keep fighting in the rest of the Empire rather than making a show of concentrating on their German possessions, there's a chance that when the dust settled they'd be left with nothing but Austria (and possibly Bohemia) while some other enterprising monarch had got up, made a pretty speech touching on all the Romantic talking points and become Emperor of the German People.But the Saxons and Bavarians blocked each other's chances of taking the prize, the Duke of Baden was never really in the running and the less said about the Hanoverians the better. If the Austrians don't see that they're losing Illyria, Italy and Hungary and that they had to concentrate on what they could still salvage, they might have left an opening for a ruthless Prussia to seize.
So there's a window. Could the Prussians have seized the prize?
It's possible. You'd need a Prussian King liberal enough to go to the Diet, and a chief minister good enough at manipulating people to get his way. Remember, that this is also the one moment when being associated with the Bonapartes wasn't necessarily a bad thing- Napoleon II was returning to Paris for the Six Months of Glory before his assassination saw the Second Republic establish itself as the dominant power west of the Rhine to this day. The Bonapartes had a powerful myth, and while the Hapsburgs have tried to write this out of history there was a sizable strain of German romanticism that thought the Emperor was a great man laid low by hubris,* not the Corsican ogre. It would have been incredibly dangerous for the Prussians to play on their past association, but it could also have got them the support of the only faction of German nationalists who didn't have a candidate at the Diet.
Unlikely? Sure. Impossible? No.
*OOC: I think that in a timeline where the Austrians 'unify' Germany with the resources of many non-Germans that there'd be a strain of Romanticism that saw the Napoleonic period as a battle between the new, vital, exciting revolutionary dynasty- and the old, familiar, archaic Hapsburgs (with their Romanov allies.) Little states having to choose which family they stood with- which side of history they'd stand with, et cetera. You can imagine the legend. It would just be one story among many, of course.