EDIT: Apparently I read the answers wrong, and the correct answer is actually the one I chose, i.e. 書こうにも.
Then my question would be focused on the difference between these two forms. I guess in this fixed usage 書こう would be the grammatical one to use, while 書きよう would be used when the speaker wants to make a proposal in general, even though both are "volitional forms"? Or is my understanding incorrect.
Question from Shin Kanzen Master N1 Grammar:
I chose 書こうにも but the answer was 書きようにも
However, there was a real question from the 1995 N1 exam that was:
which used 書こうにも
Therefore this question was quite confusing to me.
I guess a more general question would also be, is there any difference between these two forms for any word, or are they both volitional forms which are interchangeable in most situations?