You have to shoot him 3 times with C, after the first phase is done you need to go either to the left or right corner as soon as possible, take a hit and this will give you just enough I-frames to one-cycle him in the second phase. One C shot is enough for the first phase, but the one-cycle will not be as consistent.

Viable on US version as well, but is kinda tricky. Not taking a hit does not seem to save time for JP version.

Sheena trick not to make Dead Eye Joe fly on second phase.

And this is faster and bit harder version. Your head has to be placed 2 holes away from the background bench.