I know that they exist, but I am not aware of how they work. Pujie never ruled in OTL so I'm not sure if a regnal name was ever even decided for him. I know of the specific naming format for Japanese current and past emperors, but I'm not sure how it works for China. And besides, anglophones don't usually call foreigners by their proper names anyways, such as in the case of Chinese and Japanese monarchs and their titles.
It is the scholarly standard even in the west to refer to Emperors of China by either Posthumous, Temple, or Era Names. Posthumous is used in the case of dynasties from the Han through Jin to Sui, as in the name of Wen Di of Sui (Emperor Wen of Sui). This is primarily due to the fact that the tradition of Temple Names hadn't fully developed and not all emperors even had them. Temple Names are used for Emperors of the Tang, Song, and Yuan dynasties and all in between, for example, Emperor Qinzong of Song. This is primarily due to the fact that by these times the Posthumous names had become extremely long and complicated and unuseful for common discourse or scholarship, and the Emperors of the times often took numerous era names during their reigns. The Emperors of Ming and Qing are known primarily by their era names because it had become common at this point for emperors to take only one era name during their reign.
An era name is usually constructed of a poetic and auspicious statement about the Emperor's goals and ambitions for his reign and for the time, whether cultural, martial, and industrious. It can be a flowery, head-in-the-clouds Daoist or Buddhist phrase or a down-to-earth declaration of quality. It is used by the Emperor or whoever controls him as propaganda for whatever is to be achieved or maintained, or simply a declaration of the circumstances of the Emperor's rise.
If you want some really good examples of Era names, go to Wikipedia's lists of the Emperors of Tang, Song, and Yuan, as these emperors produced dozens of them and looking at them can give you an idea as to what the conventions are. Here are several good examples of different kinds of Era Names:
Hóngwǔ - Vastly Martial, used by the first Emperor of Ming
Dàoguāng - Luminous Dao, used by an Emperor of Qing
Bǎoqìng - Celebration of Treasure, used by Lizong of Song
Yǒnglì - Perpetual Calendar, used by the last claimant of Southern Ming
Yìníng - Tranquil Righteousness, used by Gong of Sui
Wénmíng - Bright Culture, used by Ruizong of Tang
Jiànzhōng - Build up the Centre, used by Dezong of Tang
Wǔdé - Martial and Virtuous, used by Gaozu of Tang
You get the idea.
As for Temple Names, they are rather easy and much more systematic. The founders of dynasties are typically reserved the "-zu" ending and typically are given either Taizu or Gaozu. All others get the "-zong" ending save for extremely exceptional Emperors within a dynasty. For example, Genghis Khan was given the Temple Name Taizu of Yuan, while Kublai Khan was given the name Shizu of Yuan for his achievements. The character at the start of the Temple Name is a characterization of the Emperor's reign (they are given posthumously), like Wǔzōng, or "Martial Ancestor", Guāngzōng or "Brilliant Ancestor", etc. Like Era Names, just look at precedent.
These names have certain conventional forms. Era names are stated as "The X Emperor", the conception being that the Emperor in question was Emperor during X Era. For example, the "Xianfeng Emperor". Temple names are often stated as-is preceded by the name of the dynasty. For example, the Hongwu Emperor of Ming would be referred to as "Ming Taizu". Shortened posthumous names can also be used, and they are typically stated in a similar way to western regnal names. For example, Emperor Wu of Liang.