The (very short!) paper [*Double points on hyperbolic surfaces*](https://doi.org/10.1090/S0002-9939-1993-1160302-7) 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. 

I think that you want to think about such curves 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.