Almost by definition, a southpaw will be a counter puncher. Also, by fighting in close, shoulder to shoulder, you surrender most of the advantage of fighting southpaw, that being the angles created. Distance, and the ability to control it, are your best friends.
I don't see the necessity to fight over every inch of the ring, or of bouncing on one's toes. Can he punch as he moves to the left- or does he just bail out going that way?( I notice that you say he moves left and then back to punch) Shorten his base and work on how he pivots and punches: being able to effectively turn to his left and punch gives him punching angles, prolongs exchanges (with your fighter in a hard place to get hit and able to punch) and should dispel the idea that he is being pushed around.