TL off the top of my head.
So have Ward somehow continue to live, and maybe consent to take a wife related to the Emperor so that they can be sure of him. Li Hongzhang his second in command can maybe convince him. Continuing to lead the EVA, his exploits rise and he and Li continue to modernize parts of China. TR hears about his fame, and in his moment of grief and after the political convention, instead of heading out to North Dakota to hunt buffalo, he gives in to despair at the Republicans supporting his enemy James G. Blaine and leaves the country, coming under the aging Ward and Li's tutelage.
There, now TR is in an official capacity in China, with ties to high ranking officials and even the emperor in a roundabout way. Ward is getting a little old by this time (mid 50s) so he's probably going to rely more on the younger TR.
I am not a Gordon fan, so I threw him in as another European Expert brought in to bolster the self-strengthening movement, but proves himself a traitor or usurper or what not. Extra twisty? Have him be an agent for the British Crown whose job is to sabotage China's gaining strength to allow for greater British (and European) influence. TR makes a name for himself by beating Gordon and well... not my TL.