A POD after Genghis Khan's birth that nonetheless produces a China equivalent in strength to the Han, Ming, or even Northern Song dynasties at their peak would have to be so incredible as to be implausible. It would require either the Jurchen Jin to conquer all of China in an implausibly short time frame (essentially impossible) or for the Song Dynasty to make major institutional changes and regain most of the north (unlikely in most ways).
But the key phrase is "Any of Han, Tang, and Northern Song at its peak." On the other hand, if a figure equivalent to Genghis Khan arose during the early years of the Han or Tang Dynasties, as the head of the Xiongnu or Tujue nomads, then China could find itself in a difficult situation. The Xiongnu nearly captured the Han Emperor in 200 BCE. The Tujue nearly reached Chang'an in 626 CE. Had a figure similar to Genghis been in power during those critical years, the newly founded Han and Tang empires could have been defeated in their formative years. However, the Han eventually learned how to survive the Xiongnu, and the Tang gained the upper hand over the Tujue soon afterwards, and kept their northern border secure.
But the Northern Song would not pose much of a threat to an alternate Genghis. The first few Jurchen rulers who ended the Northern Song were formidable leaders, but they were not Genghis's equals. If they were, Southern Song would not have survived.