As long as Austria and Russia have conflicting aims in the Balkans, the Three Emperors League won't work.
If the Ottomans win in 1876-1877 however, I could see it possibly lasting. There wouldn't be the same Balkans competition between the two as historically and the Ottomans would strong enough to be something of a counter to them.
I think you could see both nations get over their Balkan conflicts if they faced serious internal issues serious enough to refocus public attention. The Pan-Slavic movement (rivaling the Pan-German one) pushed much of the Russian positions. Austrian interest rest more on a complete lack of alternative options or theatres more than some kind of driven aim. She just wanted to avoid collapse or revolts to her Slavic minorities. As you say, I think the key to such a league working out, and even on the propensity for a war rest with the Turks. If we stick with the OTL Russo-Turkish 1877-8 war ...
Tweak German reactions to the Russian victory. Instead of Bismarck coming off as selling out Russia and doing a lot to change and wreck Russo-German feelings (Russians felt stabbed in the back after standing by during the Franco-Prussian War). Perhaps Bismarck alters his attempts to mediate and instead allows something much more like the Treaty of San Stefano to stand, thus forging renewed cooperation between the two giants. This is prior to the duel alliance, so Austria might remain outraged, but it seems little chance she would be able to find common ground with France. No one is going to formalize an alliance that only serves to push Germany and Russia even closer. A-H would have little choice but to join and at least voice her interests versus remain outside the league and see her influence trampled only further. If we are trying to push for a POD that also eventually leads to an OTL like war in Europe then perhaps the Germans back Russian dreams of the straights and the Brits start down a path of containment, scooping up what she can to encircle the league. This would be difficult to do, but I suppose a strong Japan ready to strike and distract Russia on a much larger scale in the East, coupled with some kind of Italian and Turkish buffet full of Austrian (Italian wish list) and Russian (Turkish wants) to take some kind of advantages of a Russo-German (and A-H) power. Most likely this conflict would never be started by the British anti-emperor alliance of course.
Tweak the OTL for a Turkish victory and I suppose you could see Russia turn East, but also become a bit more concerned about her internal stability in some of her minority regions. The lack of confidence influences accepting an alliance with the Germans and curtailing conflicts with the Austrians. A lingering Turkish presence in the Balkans perhaps gives the Austrians a focus on her versus conflict with Russia, opening up further cooperation for a longer lasting 3EL. The perceived threat of this 3EL eventually moves powers to form some kind of attempted containment alliance much like the one I mentioned earlier. With Turkish victory and with more generous French or British financial and military support, she presents a stronger ally than OTL, and the sum of all states in this alliance form enough of a bulwark to the 3EL that it presents at least something strong enough to avoid domination. Perhaps in a moment of internal division or strife within A-H or Russia, war does break out when some leaders sense a 'now or never' attitude toward the league.