Curious that you said that about the best way to beat a southpaw. The standard book is to get outside their right foot and avoid the left hand. I've long thought that the way to do it is to draw the left hand and counter it. The left hook, if thrown short, is a pretty good punch for this purpose, but I think that the right uppercut is a very very neglected punch in fighting southpaws. It comes down to, I think, the ability a southpaw has with his left hand. A lot of very good southpaws- maybe even most- tend to hook a bit with the left and I can see why, and there is a variety of reasons, but I think (and this is all speculative and varies from boxer to boxer) that a straighter left hand is more efficient...
Counter his left with your left, then the right. That's my thought.