I would have supported The Ottoman Empire.
I don't say this out of any love for the Turks, or their mode of government, or anything else. Instead, it would be purely down to strategic interests and geopolitics. For better or for worse, they were the ones somehow able to keep the Balkans and the Middle East from exploding into a series of awful conflicts that just entangled the Great Powers even further.
No, they weren't. Certainly not by the 19th century, when it hardly keep the peace within its own territories or even hold on to them. You seem to have forgotten that the war only happened because the British were too disgusted with the atrocities the Ottomans had used to crush a rebellion against their rule to back them up against Russia.
The Russian Empire, by contrast, sought to actively destabilize the region and force a new status quo that would see it conquer the Turkish Straits. From there, Russian fleets could act with impunity to threaten broader European commerce and trade carried through the Mediterranean Sea. Keeping them bottled up and impotent would restrict their freedom of action, so naturally backing the Turks makes sense.
Ignoring for a moment the nonsense that the minuscule Russian fleet would ever be a threat of Western interests in the Mediterranean, Russia didn't even have such objectives in the first place. Russia had tried their best to avoid war and even after that they tried their best to find accommodation with the other Great Powers. In fact, the historical evidence clearly proves how false your theories, since Russia gained nothing in the Balkans from the war and was no closer to gaining the straits after the war ended.
The best case for me would be if The Ottoman Empire was not necessarily a Great Power, but a Secondary Power, able to bear the cost of policing both of these historically unstable areas, and serving my geopolitical interests, but unable to realistically function without foreign support. Better to send Martini-Henry rifles, telegraphs, and coastal defense guns, rather than the lives of my own men.
The Balkans were unstable because of the Ottomans, who were a corrupt, oppressive and arbitrary government reigning over a mostly Christian population that they treated as second class subjects. They only became stable once they had exterminated or expelled the non-Muslim ethnicities.