I doubt it's possible, but one POD might be to have the Ilkhan Gaykhatu be successful. He had successful foreign affairs during his rule, but domestically was a failure. He introduced Chinese style paper money to Persia, which even had Chinese characters on it since the block printers were Chinese and the equipment shipped to him from China, and attempted to create a fiat currency system to deal with the Ilkhanate's economic issues. It was a disaster and contributed to his overthrow by his cousin. Gaykhatu was also a Buddhist and like many Mongols patronised Tibetan lamas.
There's also a lot of Chinese-style seals the Ilkhanate used and some of their elites had been given Chinese noble titles, like the powerful emir Chupan was given the title Duke of Yi. This is because they were the closest ally the Yuan dynasty had since both of their ruling families descended from Temujin's youngest son Tolui.
There is also an interesting event where in the 1310s Ayurbarwada Buyantu Khan of the Yuan ordered translations of several Confucian texts and a Chinese agricultural manual into Mongolian, apparently intended for ethnic Mongol bureaucrats in relation to him reviving the imperial exam. IIRC this was not unusual since there were apparently other Mongolian translations of texts of subject peoples, but let us imagine that these texts are more widely distributing and make their way into Gaykhatu, who TTL has the health of someone like Kublai Khan and lives until the 1330s. Gaykhatu and his sons and grandsons accept these texts and use them as guides for leadership and governance, and various Chinese scholars are brought to the court as advisors or bureaucrats. So now you have vague elements of Confucianism in the government too informing its political philosophy. Of course, I'd expect this sort of Sinification would have plenty of Ilkhan elites (including ethnic Persians, Turks, etc.) accepting Chinese titles.
The Ilkhans imploded due to lack of a clear successor giving a casus belli for powerful generals to fight each other, but that wouldn't be an issue here since Gaykhatu did have a grandson who was installed as Ilkhan in that chaotic period. Unfortunately, there's also the Black Death to consider which devastated Iran and the fact that a primarily Muslim region with many powerful Muslim emirs will resent a Buddhist dynasty ruling it. I doubt it's something that would survive much past the 1350s, since some conquerer be it an alt-Timur or Anatolian Turks will score a major victory and cause the local emirs to defect and start a chain reaction that brings everything down. Long term influences might be bits of Chinese technology, artistic and architectural design, and perhaps Buddhist influences on some of the Sufi orders. Confucianism might survive in diluted form in terms of how Iran perceives it relative to the world and in the political philosophy of its next rulers.
None of this would necessarily count as a truly Sinified dynasty--like for instance they won't be writing Persian or Arabic in Chinese characters or using Chinese in their bureaucracy, although maybe they would try and write Persian in the Tibetan Phagspa since the Mongols intended it would become the universal script. But it's probably one of the better possibilities for anything approaching it.