Qing was at its most vulnerable after the Second Opium War. Its army was in shambles, its treasury spent, the rivers were flooding, and by Confucian philosophy it seemed as though the dynasty had totally lost the Mandate of Heaven.
The reason Qing didn't fall, when to all analysis it should have, rests squarely at the feet of Hong Xiuquan, leader of the Taiping Rebellion. Hong, rather than being a traditional Chinese peasant rebel out to establish a new dynasty, was a madman adhering to a bizarre messianic syncretic Christianity which was palatable to neither the Chinese nor Europeans. The reason so many followed him in the first place was his charisma and the sheer desperation of the populace. Hong's rebellion, weighed down by the founder's failures, sapped the rebellious energy out of south China for decades.
The solution to this is simple. Replace Hong with a man out to establish a new dynasty and restore Han rule to China. Have him pass the civil service exams to give him the administrative know-how to manage the new state. Also, have the rebellion begin immediately after defeat in the Second Opium War.
I've actually laid out a similar scenario in this timeline:
http://www.youtube.com/watch?v=d4yNCDYepZI
