Additionally,
That it's awkward in comparison to "Whose book is that?" may have to do with structural weight. Add in "book" and the sentence's structural "weight" is no longer even:
EX: Whose is that? (even)
EX: Whose book is that? (even)
EX: Whose book is that book? (even)
EX: Whose is that
book? (odd, as in opposite of even)
acceptable 