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?
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.