I was surprised to hear some minor tonalties in the B section (around 0:52 for example) and wondered how that was being done. Sounds like the LH is playing major against the RH minor... which you don't really even notice unless you listen closely. Neat trick.
That's the appeal of diatonic button boxes - to be able to squeeze a lot of sound, and rhythm, out of a small, lightweight box. (You get minor 2nd chord on one bass button pull). Otherwise, as you said, you cobble together what you can get with right hand combinations. This is a Hohner Erica in A/D: