Traditionally, l.h. plays dots from the bass clef, r.h. those from the treble clef. And that will work here, too. There's nothing too jumpy for l.h. then. You could play the dots with stems down using l.h., but that's almost stride piano - not the easiest! The reasoning behind the stems up/down on the treble clef is that the stems up are the 'tune', while the stems down will be played slightly quieter, as accompaniment.
So, keep the hands and clefs separate, as writ, and it'll work just fine.
Incidentally, the top line (extra treble clef, so far empty), will eventually have a vocal line, which will start after the intro, perhaps more bars later than shown here.