The (very short!) paper Double points on hyperbolic surfaces by Jørgensen and Sandler gives a necessary condition for a triple point. They do not assume that the triple point is the only self-intersection. So they do not answer your question. But their techniques are directly relevant.
You should think about your curves as lying in the four-holed sphere or twice-holed torus. Then, hopefully, the extra boundary components will pair up and give the desired examples in genus two.