Can Chinese republicanism be avoided entirely? The Qing are overthrown to be replaced with another dynasty instead of the republic experiment?
You can't avoid it entirely by 1900, but you could theoretically direct it towards a more reformist, Han-Chinese nationalist path where the Son of Heaven is retained as a popular figurehead.
It is far from impossible, but a lot of things will have to happen on a same time. The monarchist reformist supporters will have to be able to mobilize their respective supporters on a right time and much better than in OTL, Qing will have to fall earlier while being totally discredited in the eyes of the Powers during the first years of the 1900s. At the same time the still-small republican movement will have to have a different set of leaders than in OTL.
And you'll have to also have a conservative faction that is willing to put Son of Heaven into a similar ceremonial position as the Japanese Emperor to support their own political goals. This group must to be strong enough to enforce their will upon the still nominally unified empire and crush the Manchu resistance. To do this, they'll need support from the military, middle-class bureaucracy of the conservative literati, and the peasant masses, as well as approval from the Powers.
As I see it, the Southern viceroys who opted to remain neutral during the Boxer Rebellion in 1900 are the only group that has the necessary influence and power to bring a new Han-Chinese dynasty around. This is also the last time in Chinese history when such a change could happen, as the fate of all latter attempts of monarchical restorations show.
And yes, as it happens this course of events is part of my current TL
