If I’ve got the hang of this, your proposition can be illustrated thus:

Take 1:

Statement: “I’m told that it is possible to engineer a happy return to a world in which secrets could be kept.”
Response: “I doubt that (it's possible to engineer…).”

In this case there is no choice to be made, so ‘doubt that’ may be used.

Take 2:

Statement: “Do you think it is possible to engineer a happy return to a world in which secrets could be kept?”
Response: “I doubt whether it is (possible to engineer …).”

In this case there is a choice implicit in the question, so ‘doubt whether’ should be used.

The use of ‘whether’ in Take 2 feels clumsy. It would feel more natural to say ‘I doubt if it is’ or more simply ‘I doubt it’. I doubt if / whether / that I would normally find a need to use ‘doubt whether’ except when choices were distinctly stated as in, for example: “I have some doubt whether he will go east or west”. Even that is ambiguous as it leaves other alternatives available such as carrying on south. “I doubt whether he will go east or west” definitely introduces further options. But then I've probably got the wrong end of the stick anyway!

I agree with Faldage; I don't believe it matters.