I don't doubt (that) that was what was he wanted.

这句话有些小语法错误，应改为这样比较好：
i don't doubt that there was what he wanted.

不能省略...引导从句呢

这里是可以省略的，that在这里做宾语

doubt that 置疑 用于肯定句
不能省略，这是doubt that 的用法
doubt whether/if (怀疑) 疑问句

可以吧...
这里的THAT起连接作用,是无意义的
这句句子是个宾语从句

我觉得直接说 I dont doubt that what was he wanted.