I think you need a POD further back to achieve an Austrian unification of Germany.
Austria didn't actually want to unify Germany under her auspices at this point. 1866 was fought primarily defensively to prevent Prussian from dissolving the Confederation and unifying Germany under Prussia. It was to preserve decentralise from which Austria could exercise soft power.
If they destroy the Prussian army at Koniggratz, the French (who had fought them only 8 years back in 1859) would be very keen to prevent too harsher settlement. It'd probably end up status quo ante bellum, maybe with minor concessions for the Austrians.
It'd stop Prussian unifying Germany, that's for sure. Unification would be very, very slow or there'd be some kind of second 1848 in the early 20th century.
But it wouldn't leave Franz as emperor of Germany.
You'd need to mess around with the Napoleonic Wars to do that. Say, have the War of the Third Coalition include Prussia and be successful, or make Napoleon dissolve Prussia in the Treaty of Tilsit. That'd leave a longer time for consolidation. Or have Joseph II successfully swap the Austrian Netherlands for Bavaria in the 18th century. Or centralise the Confederation at the congress of Vienna, to maintain the Holy Roman Empire (with Franz as emperor with for now limited power). Or even mess around with the 30 years war.
I'd suggest having 1866 be some sort of final consolidation war the that paves the way from Franz as emperor of Germany, but I'd set an earlier POD. I'd say the best bet is integrating Prussia into the Confederation of the Rhine in 1807, which weakens it beyond repair.