In Traditional Chinese these are 中華帝國 Zhōnghuá Dìguó (Chinese Empire) and 中華民族 Zhōnghuá Mínzú (Chinese Nation/Ethnicity).
I'd doubt the state would officially be called Zhōnghuá Mínzú because of its ethnic meaning, but it would definitely be used in official announcements, propaganda, etc.
My continuing thanks. Trying to navigate the Chinese language seems a little like trying to understand quantum physics to me. Well actually I probably do better with quantum physics than the Chinese language lol.
The path of the southern Chinese dynasty is pretty due to an uneasy alliance between three men. The Guangxu Emperor, Zhang Zhidong and Sun Yat-sen. These three are pretty much responsible for its creation. The key change ITTL is the Guangxu Emperor was educated in Britain (yes this really does require hooking up the TL's genitals to a truck battery to make happen, but so far it is the only time I've had to go to that extreme). This resulted in a major change in his personality from the OTL. While he kept his unrealistic expectations regarding the pace of reforms, probably even making them more extreme, he's far less under Cixi's influence and has a much better understanding of the world outside the Qing court.
As already posted, he only returned to China properly after the outbreak of the Sino-Japanese War in 1894. He had returned briefly in 1887, but at that point he wanted to stay in Britain for further education and his actions during that time very quickly convinced Cixi he was way too liberal in his outlook. So he got shipped back while Cixi tried to gain enough support to replace him. While she hadn't gained enough support to get rid of him totally by 1894, she had been able to convince the conservative elements in the Qing court he was not 'the right man for the job' as Emperor. So when he did come back and almost immediately recognised the Chinese were going to get their backsides handed to them by the Japanese and tried to open peace negotiations, she was able to stage a coup and reimpose a regency under her control. Wasn't hard for her to do that, the Emperor was totally unprepared for Qing court politicos and she easily out maneuvered him.
While the Emperor wasn't able to end Cixi's regency, the crushing Chinese defeat in the war did gain him a lot of support with the reformers in the court and allow him to largely take charge of the Qing military moderanisation after the war. During the next three years, he learnt the fine art of political intrigue and built up a solid support base mostly in southern China (particularly strong in Guangdong and Guangxi). So when the French, Germans and Russians forced a series of humiliating territorial concessions in late 1897 and early 1898, he had enough support to end the regency and take direct control of the Chinese government. This is when he plunged headfirst into reform (roughly the OTL 100 days reforms, though he got more like 200 days ITTL). As with his attempts in the OTL, he pushed way too far way too quickly. Plus his anglophillia meant he favoured British and US interests too much, freezing out the French, Germans and Russians. This allowed Cixi to stage another coup (backed by said French, Germans and Russians along with Yuan Shikai).
Cixi still couldn't get enough support to get rid of him totally yet, but she did put him under house arrest as per the OTL. This is when she started backing the Boxers. The Boxer War went pretty much as per the OTL up until the Emperor's supporters sprung him from house arrest. He went to Nanjing and was able to form a rival government with the support of the leaders of the Pact for the Mutual Defence of the Southeastern Provinces. Well except Yuan Shikai, who realised the Emperor was very much NOT keen on him after his support of Cixi's coup, so he fled to Shanxi, taking a good part of his army with him. He'd ultimately join up with Cixi when she fled to Xi'an. This when China split, with two competing governments, Cixi's in Xi'an and the Emperor's in Nanjing. The Xi'an government will eventually become the Zheng in northern China and the Nanjing government the Xianfa in southern China.
The Emperor appointed a Grand Council composed largely of Han Chinese headed by Zhang Zhidong and brought the more radical reformers such as Sun Yat-sen into his government. This alienated most of the Manchu officials, but did buy him enough wider support to survive what came next. By now the Great Powers had crushed the Boxers and occupied Beijing. Realising there was no other option, he moved to negotiate with them before Cixi could. While he managed to play the Great Powers off against each other fairly well (those three years learning the intricacies of Qing politics plus his European education paid off here), the terms imposed where harsh and humiliating for the Chinese. This cost him a lot of support in China, something the Xi'an government did not fail to capitalise on. The Xi'an still had control of a significant portion of China's new armies and launched a military campaign to regain control of China. They didn't attack the Great Powers, but they did attack the Nanjing forces starting with Hubei.
Initially the Nanjing forces held fairly well, but then Li Hongzhang, one of the Emperor's biggest supporters and commander of a large portion of the best Nanjing forces died. His successor defected to the Xi'an, handing them much of northern China along with the majority of the new armies and the two modern Chinese arsenals at Hanyang and Chengdu. From here, the Nanjing government was fighting for its life. They still had some extremely good troops from Guangdong and Guangxi, but they were stretch very thin. Plus the Europeans were very reluctant to sell guns to both sides after the Boxer War, so the Nanjing were very short of equipment. Eventually the British did agree to allow the Nanjing government to buy weapons, but this was expensive. Given what they had, the Nanjing did pretty well for awhile. Mostly due to a particularly good general on their side, helped by some infighting and incompetence on the Xi'an part. But eventually the Xi'an were able to buy off one the commanders of the troops defending their capital at Nanjing. This forced the Nanjing government to abandon a very large chunk of territory, including Nanjing and Shanghai, and brought the remaining undecided provinces over to the Xi'an.
This is where I'm up to in the TL development, with the Nanjing government having relocated to Guangzhou and looking like its about to go under. My sketch has them surviving, mostly due to Yuan Shikai staging a coup against Cixi, declaring himself Emperor and offering the Nanjing a ceasefire to allow him to consolidate his power. The narrowness of their survival forces the three major players in the Nanjing government to develop a compromise system between the radical reformers led by Sun Yat-sen who want a republic and moderates headed by Zhang Zhidong who want a constitutional monarchy, with the Guangxu Emperor playing mediator.