Okay, suggestions:

1) What is the theme you are working on? Can your character have a sub-plot related to that?

2) The other two characters are clearly going to end up in the same place - how does this third one end up there as well? Or does he not make it?

3) Instead of focusing on the human, focus on the dog in that sub-plot and make it about the animal.

4) Is the character a different archetype - e.g. the mentor - and so flits in and out of the story without a subplot of his own (Yoda in Empire Strikes Back)?

5) Is the character necessary?

Anyway, good luck!