How does the Ottoman empire taking Persia make a longer lasting Persian cultural influence? Doesn't that just drive the Turkic influence firther east?
But the thing is; all Muslim Turkish cultures were *heavily* influenced by Muslim Persian culture, and if the Ottoman Empire (of which the culture was already heavily influenced by Persian culture) expanded further eastward instead of westwards into Europe, then the inclusion of Persian territories in the Ottoman Empire would have resulted in (an even) greater Persian influence on Ottoman culture.
As for the challenge as a whole; the Persian cultural sphere already
was immense in OTL - it includeded pretty much all of Central Asia, Iran, the Mughal Empire (Persian was the language of the Mughal court, and the Urdu language is basically an amalgam of Persian and the predecessor of Hindi), good parts of the Ottoman Empire, and Persian influences exist in various Muslim cultures in other parts of the world (like Malaysia and Indonesia) as well.
And with a POD no later than 1500, I don't really see how the Persian cultural sphere can be made much bigger than it already is at this point.