I think you have the right idea, and IMO you don't even need to have them lose to Muhammad Ali completely. Have the Russians step in as the biggest champions of Ottoman territorial integrity out of the Great Powers during the leadup Oriental Crisis, arguably you could go back to the Greek War of Independence. Keeping Russia out of the Greek camp is difficult IMO, but plausible. Nicholas I is Nicholas I. I think it'd be less about having him outright side with the OE, and instead giving Russia a different lead-up to OTL's Treaty of London + Navarino.
First off, the Ottomans must not hang the Patriarch in 1821. That is, by far, one of the stupidest things Mahmud II ever did. In return, the Russian court will feel far less pressure to condemn the Ottoman Empire. Hell if anything, Gregory V whether sympathetic to the Greek rebels or not provides a diplomatic channel between the OE and the rebels as well as a very prominent Orthodox voice against rebellion for Russia's Eastern Orthodox audience. This establishes an axis for conservative Orthodoxy to align with the Ottoman Caliph regardless of the faith these liberal(more on that next) rebels profess. This will blunt the impact of Ottoman atrocities somewhat not just in Russia but throughout Europe. Straight up cartoon villain shit and an enormous self-own. This could additionally delay or even outright prevent Ioannis Kapodistrias from departing from Russia to side with the Greek rebels, denying the rebels it's most important leader, diplomat, and statesman.
Next, the Decembrist Revolt. Have it go worse than OTL for Nicholas I. Instead of being disorganized, have someone that's half-competent and also a blatant, shameless Hellenophile at the helm lead the charge on trying to force Nicholas I to accept a constitutional monarchy. In response to Russia being more pro-Ottoman, the pro-Greek liberals become more radicalized and wear their philhellenism as a badge of honor/liberal street cred. Have the ties between this Hellenophile, the Society of Friends, and the Greek rebels shine like a flaming neon sign. Presuming Kapodistrias is still in service of Russia at this time, this could send him down a different path where his views on the Greek rebellion harden for the worse. Instead of becoming supportive as a result of the death of the Patriarch, he becomes its biggest detractor in response to the coup attempt against the tsar. The Society of Friends becomes a public enemy of the Russian state. Russia switches from condemning the rebellion to outright chomping at the bit to crushing the rebels. The first true détente between Russia and the OE occurs.
Returning to that conservative axis, this gives us a nice division on ideological lines. Britain and France on one side, Russia and the Ottoman Empire on the other. The latter two now are inclined to view liberalism as a tool of Western imperialism and interference in their own internal affairs, and both will feel vulnerable after each having to face(the Ottomans still facing) a liberal revolt that received blatant support from the West. This also gives both the Russian Empire and the OE breathing room and support to carry out critical reforms(ex. the Auspicious Incident) while hashing out their areas of dispute such as the Caucasus and the Danubian Principalities.
This also gives us a lot to work with. Russia can serve as a moderating force on Ottoman atrocities while acting as the bad cop, even against Orthodox peoples, that refuse to play ball with Constantinople. The Patriarch of Constantinople can serve as a line of dialogue between the two states to figure out some acceptable compromise between the right of Orthodox peoples in the OE and Ottoman legitimacy. I think this'd also lead to a reassessment of Russian support for Serbian autonomy; in that moment in time they've got a pro-Austrian dynasty in charge of Serbia and little to show for it other than having given fuel to nationalist uprisings in their now ally. This view will only get bleaker when Congress Poland revolts and they begin to believe that nationalism anywhere is nationalism everywhere. Both states are large, multi-ethnic, and to some degree multi-cultural with an asterisk while proclaiming hegemonies over a wide range of peoples. Effectively empires that ran on faith over nationality, and sometimes not even that in the case of the OE.
So apart from the Greek revolt having a worse time of it or even sputtering outright, assuming we get an atl-Treaty of London then the Russians will absolutely oppose it. It'll be a strictly Anglo-French declaration with a lot less justification behind it if the Ottomans have been leashed off of their worst impulses between necessity, circumstance, and dumb luck. Alt-Navarino will likely still be a loss for the Ottomans, the Russians can get involved on the side of the Ottomans if they please. I don't expect much from their assistance at sea against an Anglo-French fleet. Regardless, the Ottomans will fall deeper and more willingly into Russia's orbit, seeing them as vital to the defense of Ottoman territorial integrity while shifting their focus to their navy(presuming the Greek revolt is largely under control by this point). The biggest threat to the OE will perceived as internal revolts and the ability for the Western powers to project power overseas.
End result, the Ottoman Empire is happily with the Russian sphere of influence by 1830, and OTL's defensive pact is likely a full alliance. The Ottoman Empire will send aid to the Russians during Congress Poland's revolt, while the Western powers will actively support it. This will anger Prussia and Austria while further isolating France and Britain from the rest of the continent. In turn, if/when Muhammad Ali revolts, the Russians will be inclined to more aggressively support the Ottomans while the British and French will in turn be more tolerant of Muhammad Ali's ambitions or even outright support him in potentially usurping the House of Osman.
Now to really lock in the Russo-Ottoman bromance; with Western markets more hostile to Russia while the Ottoman market is more reliable the dynamic for Russian grain exports will change. The Baltic Sea will be seen as unreliable, while the Dardanelles will be seen as unreliable. Free movement of ships is a given, but more importantly, the Ottomans will likely be the major recipient of Russian grain. This will help the Ottomans a lot demographically in this time period due to having access to cheaper food, but simultaneously make them extremely dependent on Russia such that going to war with Russia would be akin to signing off on mass starvation.
The Ottoman navy will effectively be the second(arguably even the primary) arm of the Russian navy in practice, and is likely to be stronger than OTL where it was already quite respectable. This earlier naval buildup with a reliable patron and better finances has a lot of potential to set off sparks if the French invade Algeria similar to OTL. This will further bolster the Russo-Turkish relationship. At that point, the ideological battle lines will be well and truly drawn and it'll be trivial to continue to reinforce these biases in much the same way that OTL reinforced pan-Slavism due to Russo-Turkish hostility. The two powers will be way too deep in bed with each other to decouple lest they get jumped by the West.
If/when the League of the Three Emperors is proposed, it's instead the League of the Four Emperors and includes the Ottoman sultan. Russia will be in a much stronger position politically and economically with a friendly OE whose market they have easy access to and which effectively acts as a giant outer shield for Russia proper. It's most exposed borders would be the Baltic Sea, the North European plain with the Germans and Central Asia, whereas its most secure border is the Black Sea which is kept safe by virtue of needing to go through its ally first before a foreign power could hope to attack Russia's underbelly. This would remove a lot of drive for expansion into the Balkans and Anatolia; there would simply be no need.