A: Why is John washing the car but isn’t David?
While I agree with others that (A) is ungrammatical, I don't see the problem as being whether to use "but isn't David" or "but David isn't" at the end of the question. First, "but" is not as central a coordinator as "and" is, and this can easily become apparent when ellipsis is involved, as it is here. I think that "and" works much better here than "but."
Second, instead of saying "and not David" or "and David isn't," it would be better to say "and David not." By the time "is" moves to the front of the question, beneath "why," it does so in the company of the second "is," the two verbs becoming one at that stage of the transformational derivation of the sentence. We can even see this in the full version of the sentence, without ellipsis:
Why is John washing the car and David not?
Why is John washing the car and David not washing the car?
Note that this phenomenon will only be observed with the first auxiliary verb in the stack of auxiliary verbs, if there is more than one:
Why has John been washing the car and David not been?
Why has John been washing the car and David not been washing the car?