Well, this is a piece designed for letting left and right hand fight. The basic hint for how to approach it is that the bottom line is more dense as well as legato. The top line is accentuating the bottom line.
So the main approach would be to play the left hand in a smooth legato which means that it gets first choice of fingering and good hand position. Then you see whether this allows you to intersperse the right hand. Staccato benefits from a larger approach path so I'd try crossing the right hand over the left first, shifted in a bit (the contentious notes in the left hand are all white keys, so the left hand is likely a bit more backwards). Possibly try just the top two notes for the right hand at first, but without using the thumb (which you'll need later on for the crossover).