This came up relatively recently, I think - only thing I could really come up with is a Muslim son of one of the late Byzantine-Ottoman marriages briefly holding some amount of power somewhere as a sort of nominal ‘emperor’ or claimant.
There’s some tragicomic potential in a particularly despondent Emperor, perhaps an alt!Constantine XI with the Ottomans literally battering down the gates, trying for some absolute last minute Hail Mary by converting and then immediately being done away with.
Neither really fit OP’s specifications:
The wishy washy weaselly way of doing this might be nudging things so that Islam is considered a branch of Christianity and thus Muslims are more able to integrate into imperial structures (e.g. some military strongman who adheres to the rustic provincial sect of “Islamic Christianity” making a play for the throne and displacing the prim and proper Orthodoxy)- but that would probably require changing Islam so much that it is no longer recognisably Islam?
There’s some tragicomic potential in a particularly despondent Emperor, perhaps an alt!Constantine XI with the Ottomans literally battering down the gates, trying for some absolute last minute Hail Mary by converting and then immediately being done away with.
Neither really fit OP’s specifications:
I know many argue this, but the key difference is that I am seeking an internal shift rather than an external force to achieve this.
The wishy washy weaselly way of doing this might be nudging things so that Islam is considered a branch of Christianity and thus Muslims are more able to integrate into imperial structures (e.g. some military strongman who adheres to the rustic provincial sect of “Islamic Christianity” making a play for the throne and displacing the prim and proper Orthodoxy)- but that would probably require changing Islam so much that it is no longer recognisably Islam?