In the question, you compare the way Sun Wu legitimized its rule of the south to the way that Cao Wei and Shu Han did. Unfortunately, this tends to obscure the fundamental differences between those two states and Sun Wu, which makes it difficult to discuss this topic correctly.

And to understand why, we need to consider why each state needed to be legitimate, and to who. Cao Wei ruled nearly the entire historical Huaxia heartland, one that had been ruled by the Han for nearly 400 years. Shu Han was led by northern immigrants who held similar values of Han restoration. But Eastern Wu? It was a state headed by a southern rural soldier clan, with an entourage of mostly southerners, who governed over southern elites. While many Wu subjects from farther above the Yangtze may have had interests to some extent shared by northern elites, the vast majority of local magnates in the territory had little interest in “reuniting” China. These were people who had grown rich lording over wealthy mercantile cities like Jiaozhou or Nanhai, or rural tribal lords still about as Sincized as the day the Chinese first arrived. As you know, Shu Han and Cao Wei therefore saw legitimacy as all about being as continuous with the predicated legitimacy of the Han Dynasty as possible. The worldview they created was one of worthy rump states within a fractured whole. But Sun Wu, having an entirely different set of priorities, saw differently. In their mind, declaring a claim to the empire didn’t make them legitimate because they were the last deserving rulers in a shattered civilization, but rather made them legitimate by moving the center of that civilization to their homeland.

Yes, the Wu government still paid lip service to imperial unity. Satisfying the handful of Northern irredentists like Zhuge Ke, technically the Sun clan deserved the Mandate more for their virtuous conduct and benevolent rule, in contrast to Cao “nothing-happened-in-Xu-Province” Wei and Shu Han. But the focus of imperial legitimization strategies was almost entirely on emphasizing a complete shift of the center of civilization to the south.

The religion, diplomacy, and culture of the Eastern Wu court all changed to reflect these more southern priorities. Sun Quan was, until the very end of his life, extremely lax with building the proper temples and performing the proper rituals to appease the Mandate of Heaven. Instead, he was supported by an imperial cult to the traditional southern deity Jiang Ziwen, which shared many traits to those frequently displayed by the Sun clan, like excesses of bravado and pleasure seeking.

Towards foreign relations, Sun Wu played ball with Shu and Wei, but also sent expeditions across the East China Sea, visiting Linyi (central Vietnam), Funan (Cambodia), Hainan, Yizhou (Taiwan?), the Ryukyus, and Gongsun-ruled Korea. In many respects here, parallels can be drawn to the older southern empire of Nanyue, which employed enfeoffment of foreign lords as a frequent diplomatic tool, though Wu was rather less successful with its attempt at this in Korea.

Finally, within the court culture of the state, the Wu language, likely not Sino-Tibetan at the time, became dominant in court. Sun Quan encouraged a blossoming of art, philosophy, and the sciences, developing a uniquely southern kind of sophistication. In fact, he replaced the Han Dynasty’s traditional sifen calendar, one used by the other two states (who I emphasize again, cared a lot more about continuity with the Han than Wu ever did), with the qianxiang calendar, which had been developed by a man from the region of Wu and was considered more efficient.

So to sum up, much of the Wu elite never saw themselves as having anything it needed to prove to the rest of China. Legitimization is for satisfying the people who run your country, not those who don’t. Wu made nominal claims to the throne of the entire country, but much of their claim focused not on integrating the local into the imperial, but rather on converting the local into the imperial.

I'm going to come at this from a different angle to the interesting answer already provided by u/RiceEatingSavage

I should, as said elsewhere, mention I not have had the fortune of reading the Chittick book yet. It might be a case of disagreeing on the nuance/balance here between local and national plus how much Wu drew upon Han as well but feel the need to explain why I'm coming at it from another angle.

There is a much needed and welcome focus in recent years from figures like Chittick, Farmer and Tian on cultures and literature outside of the north. Wu's efforts (via diplomacy and conquest) in the south can be assumed to be very much to support their efforts in the north and viewed from that perspective, Chittick trying to re-frame how we view this is important and sounds intriguing (as was his manifesto). About the need to avoid homogenizing the culture or reducing it to a north-south binary, to write more regional histories (as was starting to happen during the three kingdoms with Wu scholars having an interest in geography), of how different cultures were, about the use of language when describing them.

Where, when discussing Eastern Wu, I disagree is the idea that, bar a few northern families, they didn't care about imperial unity. At least at court level (whereas I think RiceEatingSavage was more concentrating on the magnate level). The lands Eastern Wu controlled did have their own culture, beliefs and language but that had been accepted by the Han, that places had their own characteristics and people were shaped by the lands they lived in. The Sun and their followers did tap into and draw upon the existing local beliefs and customs very successfully, which is part of Sun Quan's success as a warlord.

But Wu's court drew upon a shared history of the Han via works like Xie Cheng and Wei Zhao, they boasted of connections with the rest of China when suitable, and collected works from other factions (or wrote about them). They drew upon works like the calendar (by northern scholars as far as I understand it) and the Classics, Wu's ranks and administrative systems were based on what had gone on before. It put major political (including forgeries, works like the Caoman Zhuan as well as diplomacy) and military efforts into expanding northwards even if, quite often, the large scale incursions across the Yangtze failed. The discussions within the court were not about independence in the sense of keeping clear of the other states but about unification under the Sun banner.

They were of course, also happy to write about why the south was superior culturally to the north, engaging in cultural as well as military and political battles with the north. So in terms of seeking to also shift the cultural legitimacy south, I would agree. But as part of a legitimization strategy as superior over northern scholars of that time, while still drawing upon Han legacy, rather than just distancing themselves.

So onto the general question

When you talked about the fall of the Han, most sides took one of two courses, tweaked to suit their circumstances. If an ally to the Han controller then you have likely been given rank (even if not as much as one might want) to confirm your legitimacy over the lands you hold and to raise troops. You have personally been endorsed as inspector/governor of the province and even a General by the Son of Heaven and so are the clearly rightful figure maintaining order in the area till things settle down, recognized for your now entirely legal and legitimate service. Such a good Han loyalist you are and how good of you to attack those treacherous disloyal rebels whose lands you don't currently control.

If you were fighting the controller of the Han then the controller of the Han was abusing his power and the Emperor needed saving. Ideally one would have an ally who would nominate you for the promotion you wanted and you would do likewise for him to add legitimacy to claims of rank. With the belief that, in the happy idea where the Emperor was freed from the evil controller, he would confirm the ranks (and probably more if you were the one who "rescued" the Emperor). Such a good Han loyalist you are, how good of you to fight rebels and the usurping controller.

The Suns played that system well, Sun Ce's revolt against Yuan Shu saw Cao Cao confirm his position in the south and military authority (after some negotiation with Wang Pu), and Cao Cao confirmed Sun Quan's position so the Han had granted the Sun's control of their base then in 219, confirmed Sun Quan's position in Jing. While becoming King under Cao Pi was a humiliation as forced to accept a new sovereign, it was still Sun Quan being ennobled and bought Sun Quan time while giving him a rank he could use. When against Cao Cao, Liu Bei recommended Sun Quan as Governor of Xu and the prestigious rank of Generals of Chariots and Cavalry.

Then we come to the three claimants for Emperor and yes, two of them claimed via much via the Han. Heaven seemed to shine omens on just about every ambitious warlord who asked, officials with utterly random spontaneity all urged their modest masters to ascend to the role. Cao Wei had built up an administration within the Han for a while, they could point to controlling traditional heartlands including the capitals of the Han, of their scholarly qualities of the rulers and the court, and had an interest in ritual (including Gaotang Long working on the calendars). However, their big claim is the undisputed Han Emperor Xian had personally abdicated, recognizing the mandate had passed away from the Han to the loyal Cao's who had done so much. A carefully controlled process with Cao Pi rejecting the throne more than once. This was rather a useful precedent for the Sima's who could copy the abdication process even if the regicide of Cao Mao shows the process getting a little bit botched along the way

When news reached Liu Bei in the west, he acted as if the Emperor had been murdered and went into mourning. While only a frontier province, one that was not often seen in a good light, and one of the smallest of the states, it could use the prophecy traditions of the area (even if Du Qiong and Qiao Zhou then used word play and prophecy against the regime later, wasn't just Eastern Wu magnates who didn't entirely get behind the court line) and... well the Liu name. Last Liu standing, Cao's usurpation was illegitimate and a sign of the Cao family's wickedness and the Han remained the sole legitimate Empire so the Liu's continued on the Han's mandate, the throne passing from Xian to Liu Bei (then onto Liu Shan).

So we come to the last to declare, that Sun Quan and Eastern Wu after Sun Quan had manoeuvred through the fallout from his seizure of Jing in 219 and by 229 were ready. There had been a lot of work done, Lu Dai had ended the eroding Shi hold in Jiao (and Vietnam) so the Suns were uncontested in China's south. Sun Quan had spent time in Jing to secure it and establishing a second capital at Wuchang. He had, as a King, declared his own Reign Title, he had adopted his own calendar that, as has been mentioned broke from the Han one, using the work of leading Han astronomers and mathematicians Liu Hong of Taishan and Cai Yong of Chenliu, figures like Chen Hua had been justifying heaven's favour being with Sun Quan to rival courts.

There had been discussions as far back as 223 as to taking the final step but now was good. Shu-Han had been weakened when Liu Bei's campaign over the fall of Jing went horribly wrong, Wei's southern forces had taken a heavy defeat the year before and were occupied with Zhuge Liang's recent campaigns. As the second biggest power in the land, with a court of culture and wealth (as RiceEatingSavage mentioned), it had become time to also be an Emperor and claim the mandate.

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