If Austria wins at Koniggratz and manages to keep France out, it would be very likely that it would try to beat Prussia for good. The Austro-Prussian war was the finale for a century of Austrian and Prussian rivalry. This war was fought to decide who unites Germany. Austria may or may not be capable of continuing after Koniggratz but that's where Russia could come in. Austria could promise Posen (maybe even Galicia in order to have the full Kingdom of Poland?) to Russia, as they were very much interested in it after the Napoleonic wars. AFAIK Austria IOTL was not willing to let go of her Transleithanian possessions and join the greater German solution because that Germany would have been dominated by Prussia and its population (16 million Prussians vs 12 million Austrians). However, without Silesia and Posen the odds would be in favor of Austria in a united Germany (15 million Austrians vs 12 million Prussians). Most of Germany was already on Austrias side IOTL, an Austrian victory would put Prussia back in its place as a second rate great power.
Color me skeptical given how bad Russo-Austrian mistrust was at this time. Even if Russia won't intervene to actively help Prussia, it's far, far more likely to prop up, ally with, and try to puppetize rump Prussia then try to feast on it.
*If* the Russians were to betray the Prussians and collude with the Austrians, the bribe would have to be much bigger than Posen. All of East Prussia and West Prussia (the lands east of the Confederation border) and all Galicia and Bukovina would be required too. Even then, hardly anybody would think this is in Russia's interest.