Maybe if you make the empire weak in a time when religion still matters and the sultan is still the supreme ruler?
Then perhaps in one of the succession feuds you could have one son getting support from christians in exchange for conversion.
It wouldn't be very clean though to put it mildly...
There tends to be a persistent idea that the Sultans were absolutist monarchs, but the reality was this is far from the truth. Western monarchs had the divine right to rule, but in Islam, while a "divine right" was there, you forfeit it by being incompetent, insane, disabled, senile, cowardly, and most especially, not that this ever happened, apostate. There were no regencies in Islam. If you were not fit to rule, you didn't. If your father died when you were too young to assume the throne, an older male of the dynasty got it.
A Sultan had no ability to convert the empire to another religion. If the Pope suddenly converted to Islam and declared that all Catholics were now Muslims too, would that work?
Support from abroad would just make it worse - not only would he be an apostate, he'd also be a tool of infidel powers and a traitor.
By converting, he would in the eyes of everyone utterly and irrevocably forfeit his right to rule, not to mention his life.