Ras would win, easily. Nolan's Ras is basically Batman without any moral issues towards killing criminals, or anybody else for that matter, that doesn't agree with his philosophy. And that's the catch:
It isn't so much that the Joker dominates Batman physically, he gets to him psychologically. Batman had multiple chances to end the Jokers reign of terror simply by killing him, but because he can't do that morally, he allows him to continue, which in a weird way, makes him kind of responsible. And that's the beauty of their relationship and the interesting part of the dichotomy of Batman's crusade.
Ras, on the other hand, would have no such problems killing the Joker the first opportunity he gets. Stick him on the Batpod instead of Bruce Wayne, and the movie would've been over right there...