(a) Je vais faire faire la vaisselle à Pierre.

does work as is in French. Those two faire back to back might seem strange at first, but it is correct, and we are used to it, so no problem.

(b) Je vais faire Pierre faire la vaisselle.

That one is definitely broken. That Pierre between the two faire should not be there, although if you think about it, it’s logical in a sense.

(c) Je vais laisser faire la vaisselle à Pierre.

Is correct – nothing special here, maybe a tad cumbersome, maybe it’s just me – I would prefer that à Pierre to not be so far at the end of the sentence. But it is correct.

(d) Je vais laisser Pierre faire la vaisselle.

That one is correct, and probably the one I would prefer. It sounds best among those four alternatives, IMHO.