B whether为正解
he had his doubts表示他还是有疑问的,需要用whether“是否”
对比：There was no doubt that they would give him another chance.
呵呵 答案出错了 这句话是比较明显的 有疑问必须有whether
下面提供朗文字典里的例句你参考一下：
1. There's some doubt whether John will come on time.
2. There's no doubt that he'll come.