Having both Selim III and Mahmud escape assassins, as well as having Mustafa IV avoid getting arrested and later executed, would probably help well.
Have Mustafa Bayrakdar succeed at rescuing Selim III and Mahmud, but have Mustafa IV's own men manage to brutalize Bayrakdar's rebel forces, killing him too in the process. From there, one can assume whom would have whom on which side - the Janissaries and more conservative forces within the Empire would side with Mustafa IV, whilst the more reform-sympathetic ayans, pashas, viziers, beylerbeys, etc. would side with Selim III and Mahmud.
Well, one thing is for certain, the European Powers will definitively intervene on the conflict, and maybe do some land grabbing on the edges of the empire (Russians may try their luck in Armenia, Brits in the Levant and Arabia, Austrians and the French on the Balkans, maybe even Persia decides to enter the fun) while other parts rise in rebellion (an earlier Greek one maybe). the Palace Coups becoming a Civil War would most likely cause the fall of the Ottomans an century earlier than OTL
I feel as though that point may be counterproductive to the request made to have it last into the 1810s, since foreign intervention might prompt attempts at compromise from both sides.
Speaking of, Russia had already gone to war the year prior with the Ottomans, and by the time of the Palace Coups, had dragged the Serbs into the conflict, who themselves were undergoing the First Serbian Uprising (renegading Ičko's Peace at the
Belgrade Assembly in early 1807, after hearing Russia's offer, and by July [a month after the first coup] had signed an alliance proper). Depending on how much success the Russians have by a TTL Treaty of Tilsit, it might trigger attempts at compromise from both sides.
A great way for Napoleon to get Austria and Russia on his side.
Perhaps, though not as simple as that - given that the Ottomans in early 1806 had formally opted for an alliance with France against the Russians and the British, and the actions of that would in turn lead to the aforementioned Russo-Turkish War, as well as the British Alexandria expedition of March 1807. Mustafa even persisted with the alliance, and had sent an ambassador to work out the details of it. Given that the Treaty of Tilsit resulted in the Russians signing an armistice, which they temporarily kept to whilst transferring forces southward, Napoleon may have been keen to maintain the alliance.
You would ultimately need to drive a wedge between the two sides of the civil war deep enough that not even the Russians in the north nor the British in Alexandria could force a compromise and end to the civil war. That itself would likely prompt Napoleon to give up on the Ottomans, and allow the Russians to finish their war with them, resulting in not only its end by 1808, and having received gains, but also an expanded and independent Serbia that likely would've been made to align with Napoleon.