Jason Bourne is almost superhuman in terms of mental/physical skill, yet he manages to be shot on the back twice by a supposedly not-as-well-trained dictator. This is a guy who noticed that Bourne spared his life, so it doesn't make much sense that he proceeded to shoot him, or that he even managed such a feat. Is there some explanation that I missed, or is it a shortcut/oversight by the writers?
I don't think it's a shortcut or oversight. Bourne's skills are depicted as superhuman, but even he can't dodge or outrun bullets fired at point blank range. If all had gone according to plan he would have killed the dictator and walked out. Once Bourne made the decision not to kill his target he was in an entirely different situation. Instead of walking away from a dead body, he was running away from someone who was armed, scared, angry, and alert.