Switch logic circuit
Functionally, you need three 3PST switches and a stepper.  Electrically, the stepper can be a CD4013 wired as a switch debouncer plus 2-bit counter.  With 1x4 analog muxes, this gives four states in continuous rotation: all off, position 1, position 2, position 3, repeat.
1 - CD4013 dual D flipflop
2 - CD4052 analog muxes (three switches plus an unused spare)
1 - SPST switch
1 - resistor, debouncing
1 - capacitor, debouncing
3 - capacitors, decoupling