Is there an anime based on Genghis Khan or Kublai Khan yet?
Or an anime based on an ancient Chinese emperor/general that isn't from the Three Kingdoms era?
Kingdom was the most recent one, but you're much better off reading the manga for that.
12 kingdoms is chinese as fuck. And it's actually pretty good.
Wh-what about Genghis-kun?
To the Ends of the Earth and the Sea
There's a manga, Shut Hell, featuring Genghis Khan, with a kinda genderbended berserker MC.
There's several manga about him. Go learn Japanese if you want it so much.