I find the primary difficulty is a you've said the stalemate created by each fighters jabbing arm being on the same side. The advantage enjoyed by southpaws in this respect is based upon expierience. They have faced a lot more conventional fighters than the orthodox guy has faced southpaws. In short they've been hgere before and know what to do.

So get as much southpaw expierience as you can early.
For me I find the jab is a great weapon against the southpaw If you can get the southpaw circling left (perhaps away from your right?) and then get your foot outside his right you will find various ways to attack....the jab outside a shift of the feet and a jab inside....a low jab...an uppercut.

It'll all come with expierience.