It is quite true that Russia and Britain would have very little motive and likelihood to intervene and bail out Austria. Napoleon III, however, played a complex game and expected to come out the mediator to enforce a compromise peace between Austria and Prussia-Italy that would weaken no side too much, give him some territorial gains, and reaffirm France as the hegemon of the continent. He started the war as a philo-Italian-Prussian, not really expecting the Prussian army to win so much and easily. So if Bismarck wins too much, he could switch sides. And Bismarck would not certainly gamble his luck in a two-front war, especially not with Austria not being completely defeated with an intact army in Italy, which making an advance on Vienna not wholly safe, so he would bow down to a compromise. This is the main problem with increasing Prussian-Italian gains in 1866 simply by making Prussia bolder after Sadowa.
This is why I used a two-step PoD for my 1866 TL: first, by making Italy make a better performance in the war, Austria is still wholly gutted without increasing the strategic burden on Prussia. If the Austrian army of Italy is destroyed at Custoza, and the Austrian fleet at Lissa, the Prussians can march in Vienna and/or making greater claims on the peace table without fearing an Austrian counterattack with the intact army of Italy. Moreover, a successful Italy would surely make rather greater requests at the peace table, Trento and Kustenland if not Dalmatia too, which would easily motivate the Prussian elites to increase their own, to all of Bohemia-Moravia and Saxony, and overrule Bismarck on this.
The threat of French intervention is still a very relevant issue, even a completely victorious Italo-Prussian alliance in the East would not likely gamble a two-front war with France. Remember, almost everyone in 1866 Europe deemed France the most powerful nation in the continent, and Berlin and Florence likely would not risk a war with it unless they can give it the full undivided weight of theirt military power. So a compromise peace that nets the Sudetenland and Saxony to Prussia, and Trento and Gorizia-Gradisca to Italy is the most likely outcome. It would not weaken Austria so much that France would care to veto it. Nonetheless, if all Austrian armies are destroyed, risking a two-front war becomes slightly more plausible for Prussia-Italy, if the least likely option, since they would only have to leave a limtied amount of troops to garrison occupied Austria and could shift the rest back West. Of course, France could exploit the interval to break out in the valuable Rhineland and northern Italy, Prusso-Italian reserves may or may not be able to keep them out. Fearing a French breakout is the main reason why Bismarck and his Italian allies would not likely gamble a war.
Of course, in the medium term, losing the 1866 war even worse than OTL almost surely sends the Habsburg empire on a death spiral within a decade. After France is beaten down in 1870, Berlin and Rome shall have no further obstacle to do what they want of the Habsburg heritage and can safely pick up the choice pieces of Austria when it falls down of its own. Russia can be bought with Krakow and Galicia, and Britain by giving diplomatic support to its efforts to limit Russian expansionism in the Balkans.
Alternatively, if you wish to save Austria nonetheless, have Francis Joseph make a devil's pact with Russia, he accepts to become a Russian vassal and give Russia a totally free rein in the Balkans, in exchange for a Russian military intervention to save its throne as in 1848. Quite likely, this ends up into an Anglo-German-Italian-Ottoman CPs vs. an Austro-Franco-Russian Entente.