Mesopotamia was predominantly Semitic (Aramean and Arabic) not Iranian, Christian presence beyond the Zagros wasn't very significant beyond some trading communities in Sogdiana, the Iranian plateau being predominantly Mazdean with Hindu-Buddhistic communities in Afghanistan and Pakistan. Conversion of a Sasanian Shah is very unlikely in how the Empire was structured and legitimized by the Orthodox Zoroastrian religion, a situation similar to France under the Wars of Religion, even though Henry of Navarre "won" Paris was worth a mass in the end, any Shahanshah that is Christian or sympathetic to it will conform to the traditional faith the wurzugan expect him to be, or else he'll meet the same fate as Yazdegerd I.Christianity was quite popular among the Iranian lower classes (and even some sympathisers in the nobility), particularly in its political and economic heartlands in Mesopotamia. Sassanid rulers were generally quite happy to give it patronage, particularly the Nestorians who were disconnected from Rome who regarded them as heretics, and there were plenty of Sassanid Shahs who were very sympathetic to it and may have even been crypto-Christians. If the Arab conquests are prevented somehow, either by giving the Iranians a stronger position or butterflying the population boom in the penninsula that preceded it, then it's likely that a later Shah might embrace Christianity openly further down the line, probably some form of Nestorianism so they can keep up the pissing contest with Rome.
An alternative would the Sasanians being deposed by Christian noble family, but that is more complicated than you may think, Bahram Chobin and the civil wars of the 620s shows that while the Parthian dynasties were okay in working together to rein the royal power of the Sasanians, they weren't so supportive when one of them was suddenly in charge, Bahram Chobin's propaganda of restoring the Arsacid dynasty and about Pahlav brotherhood flew on the heads of the Ispahbudhans and (Christian) Armenians that still fought for Khosrow anyway, without the religious legitimacy that still reigns in most of the Iranian nobility the Christian Shahanshah will need to get his support (read, troops) elsewhere, either in Rome or the Arabs, so quite a shitfest.
The easiest route for me is, imo, avoiding the Sasanians and Kerdit altogether and keeping the Arsacids in charge, although not much is know about Parthian kingship it certainly wasn't as sacral as the Sasanian one, offering the possibility of a Christian Arsacid to establish a state religion with much less fuzz.