Would it really be that tough?
Yes.
I would think that if Britain and France explicitly told the Sultan that they were going to abandon him (maybe due to some resurgence of Hellenophilia, or maybe because they really really want Egypt) the Sultan would have little choice but to align himself with Russia and sign another Hunkar Iskelesi. Realpolitik and all that.
The consistent goal of British foreign policy throughout most of the 19th century was to prevent Russia from acquiring the Turkish Straits; the consistent goal of the Russian Empire's foreign policy throughout most of the 19th century and until the end of its existence was to acquire the Turkish Straits. France, until the Crimean War, was on the same side as Britain in this regard; when France lost the Franco-Prussian War, the British Prime Minister's greatest concern was that France would no longer be able to help Britain to contain Russia. The Ottoman Empire was a consistent ally of Britain until the early twentieth century, when Britain, mostly because of paranoia about Germany (though this is a disputed topic), cut it loose.
I do not suggest that Britain will stay so friendly with the Ottomans out of sentiment, but rather out of national interests; it is in Britain's interest to keep Russia's expansion contained because Russia is the single greatest threat to Britain's Far Eastern possessions in the 19th century, and the power of the Russian Empire was grossly overestimated in the late 19th and early 20th century so the British saw an even larger threat than the still-major one which existed.
I cannot think of anything that could
possibly make Britain turn fully against the Ottoman Empire. Even if Britain was eager to take Egypt
and Greece
and the entirety of the Balkans, it would still be preferable (in the Ottomans' eyes) to Russia, which saw the Ottoman Empire as a natural area of expansion, aimed to take the Ottomans' most valuable territory and sought to turn the Ottomans' very capital, Constantinople/Istanbul, into a Russian city.
I can't see Russia wanting to dismantle an Ottoman Empire that was aligned with it.
I can. As soon as the Ottomans had outlived their usefulness (in this hypothetical Russo-Ottoman alliance) they would be doomed. Even in the scenario that you have described, the Russians would only be allies of convenience to the Ottomans; once the Anglo-French threat was ended the Ottomans would be on the Tsar's menu.
And this scenario is unlikely, because Britain is unlikely to discard its allegiance to the Ottomans unless it thinks that the Ottomans are becoming a puppet of one of Britain's powerful rivals… in which case the Ottomans will have another powerful ally (IOTL, Imperial Germany) to protect them from Russia.
A Russo-Ottoman alliance is actually
less realistic than an Austro-Russian alliance or a Franco-Prussian alliance; either of those could have occurred had history gone slightly differently in the 19th century, but any
permanent Russo-Ottoman alliance (as opposed to the alliance of convenience described above) would require a completely different development of Russia's self-image and policy goals, long before that time.
As for the Habsburgs, see my comment above about Britain; it applies equally well here. Even if the Habsburgs wanted to turn all the Balkans, including Greece, into their puppets, they would still be a more attractive option to the Ottomans than Russia would be.