"Do you have any kind of bowl?"
Translation:Ĉu vi havas ian bovlon?
Okay, I accept what you said about the accusative, but wouldn't "ian" mean "some kind" and "ian ajn" mean "any kind"?
Well, if you say a similar sentence in affirmative, would it be "He has some kind of bowl" or "He has any kind of bowl"? I'm not a native English speaker, but the second sentence seems odd to me. Anyway, I know ia means "some" and can really mean "any" in interrogative sentences, but you need ajn when it means "...at all".
I'm no expert, but I would say that, yes, "ian ajn" should be an acceptable answer. I found some examples:
According to Lernu.net: any kind of = ia ajn
According to ESPDIC: ia ajn = any kind of; ajn = any kind of/whatsoever/arbitrary; sen ajna = without the least/without any kind of