I personally would leave the 'a' out of the sentence and say 'what kind of bird is that?'

guys i agree with you cowboy..

I agree with Michelle, but it depends on the context because it can also be used as you have said it to show a disbelief or doubt about something's integrity. a) This bird has two heads! b) A bird with two heads? What kind of a bird is that? (asked incredulously)