What do you do when he runs out?
Odds on is that he dives out to the left and you continue to turn him in that direction so he completes a circle and then go a fair distance away to represent to the fence.
The fact that he runs out to the left has nothing to do with you being weaker on your left side, after all, you would correct by being stronger with your right rein to keep him in. OK you do need to use your left leg more too but, the right rein is your means of correction.
so, when he runs out do not circle him left instead pull him up hard with your right rein and turn him back to the fence. If the fence is no bigger than 2'6" he only needs a couple of strides of trot to get over it. It might not be pretty but better than allowing him to run out. If you need to represent from further make sure you do so from the right rein and not the left. Also keep a stronger feel on the right rein.
Learn to carry a whip, I feel naked if I am whipless! I rarely ever use it but it is there if I need it.