What you propose is the correct form.
In the given French sentence, something is missing to justify the use of "pourquoi" in one word, like: "mais pourquoi faire autant de bruit ?" (but why making so much noise?)
-pourquoi ? = why? = for what/which reason?
-pour quoi ? = what for? = to what/which purpose?
I think you would need to add "le" or "ca" for it to be "why do it"? (Pourquoi faire ca?)
"Pourquoi" is one word meaning "why?" "Pour quoi" is two words meaning "what for?" That's why we were having the discussion above...
I wrote "To what end" I guess that's too obscure. I've gotten this one wrong about 4 times in a row now.
bother=faire??? bother=ennuyer, gêner, déranger, incommoder, embêter, importuner, contrarier, s'efforcer, se donner la peine. HELP ME!!! AU SECOURS, SOS
Both expressions are idiomatic, but they don't quite match in my opinion.
You would say "but why bother?" about something that you judge useless or vain, something you don't care about. In French, we would say "mais pourquoi s'en faire ?" (lit: why bother?), which in a longer version, means "mais pourquoi se faire du souci ?" - both meaning that you don't care about it.
You would say "mais pour quoi faire ?" to inquire about the purpose of doing something, without any pre judgement about the relevance of that purpose. In English, a translation could be: "but to what purpose?" or "but why doing so?"
I hesitate to try and improve on anything Sitesurf says here, she is always such a useful source; but [as a native English speaker] the phrase 'but why doing so?' isn't quite right; possibly ' but why are you doing so? ', although this is rather stilted. I would use 'but why?' in a neutral tone of voice ^^.........just saying, please don't shoot me down.
Please never hesitate when it comes to correcting me! I also intend to improve my English, so you are most welcome. Thank you for that.