Yes, good luck getting the Turkmen population of the OE to understand anything Constantinople/Istanbul tells you in that language. I created an alagous type language for my own TL and that it spoken by the court, the officials and for record keeping purposes because of all the various languages spoken in Alt-Spain. The actual people themselves can read it after a fashion and the basics of Latin are taught in settlements of about 500 or more. It's only with the beginning of printing that literacy going to move beyond 50% having even the basics.
One thing I'm not clear on, when did Persian not become a prestige language in the OE?
I guess we are saying the same thing. I don't know Turkmen, but I'm pretty sure any Turkmen tribeman would have had trouble in understanding the persianized language usually spoken at court (where Serbian and Bulgarian were widespread though, due to Balkan recruitment of many courtmen of the imperial serviceO, plenty of arabic loanwords (the grammar was not very different I think). I suppose that every tribe or rural community had a somwhat cultivated shaykh able to speak the language of the powerful, or at least to understand it.
Persian was a prestige language for the Anatolian Turks from the very beginnings of the Rum Seljuk sultanate, since Seljuks, though being Turks, came in Anatolia from Persia and their court service was already literate in Persian. This cultural prestige never ended throughout the whole history of the OE, although I think the last century might had seen a relative decline in favor of Arabic and Western languages. AFAIK, the Persian cultural imprinting remained very strong in the non-Arab provinces of the OE as well as in Iraq (obviously) all the time.
Poets wrote verses in Persian even in the Western Balkans.