I was going through the creature's from John carter and other than Bantha and White apes,I am not sure how many other dangerous monsters are present.So to make the Forest of Monsters more dangerous for it's name sake some of the creature's from Avatar would be in the forest like Thanator,Viperwolves...etc.
The Tharks will use Direhorse like the Navi used them,but without the bonds and the direhorses are smaller to match humans or Tharks size.Also there will be Banshee's present to inner region of forest.
I remember they do have flying mounts called Malagor is a very large bird native to the planet Barsoom. It is thought to be extinct by the general population.
Please do comment if this is good or not,since I remember James Cameron talking about taking inspiration from John Carter.