Without rambling on too long about the background, suffice to say the Austrian Empire was not in an enviable position near the end of 1866. Faced with a pair of disastrous wars against Prussia and the French-Piedmontese respectively, the lose of its influence in Italy and Germany as her allies/clients fell to her rivals, without allies after her refusal to come to Russia's aid in the Crimean War, a looming financial crisis, and simmering resentment on the part of her minor ethnicities there was a very real chance of the Hapsburg dynasty tumbling from Great Power status if this unbearable pressure continued much longer. It was only by reaching a compromise with Hungary and the (relatively) generous terms offered by Prussia following their brief Fraternal War that Vienna was able to find the breathing room to recover. However, this wasn't the only possibility: riding on a string of victories, King Wilhelm I had wanted to continue the Austro-Prussian war until he could have a triumphal march through Vienna, potentially annexing Austria or at least large parts of the Empire. Only Bismark's stringent resistance; up to and including a threat to resign, on the basis of keeping Germany acceptable in the eyes of the world and limiting Austrian desires for revenge allowed the war to be surprisingly short as it was; helping lead to Germany's unique unification and Federal structure later on.
While Wilhelm I completely ignoring the advise of as prestigious a man as Bismark is fairly unlikely... at least much less likely than his grandson, what if King Wilhelm had taken advantage of his chancellor's penchant for boosting/maintaining Prussia's reputation as a power broker abroad to reach a compromise with similar results to the Russian victory in the Russo-Turkish war of 1877: namely, that while Bismark would stay on while Whilhelm finished the war and created a provisional treaty, Prussia would partake in an international conference with the other Powers to discuss the future of the Hapsburg domains. For the sake of making this more believable/likely to be accepted by Vienna, let's say as the war drags on and Austria tries to raise taxes/conscripts to rebuild its shattered army, this triggers a crisis/revolts in Hungary Bohemia and smaller, less organized outbreaks of violence throughout the Empire as the local notables demand more autonomy, if not outright indepence, and resist Austrian attempts to take their men and materials to fuel what they see as a hopeless war over "German interests"
A few points to consider.
-Who gets to participate? Great Britain, France, and Russia are obvious, but what about Sardinia-Piedmont and the Ottoman Empire? Other German states like, say, Barvaria? Are their representatives of the Crown of St. Stephen and St. Wenceslas?
-What kinds of proposed divisions can we expect to see? I imagine they won't exactly decided mainly based on ethnic line, but partially on the interests the present parties have in the "Controlled breakdown". I personally don't think we'd see complete dissolution, but its a possability.
-With Austria broken down as a Great Power, how are British interests in the Ottomans affected? After all, there isen't any other potential check on Russian expansion; are the Brits going to be more dedicated to building up/propping up the Sultanate?
-How does this impact German unification?