Intimidate refers to the skill of getting people who do not like you and who do not agree with you to do what you want them to do anyway.
You can recognize the use of the Intimidate skill by the fact that it has a threat in it - that is the proposition stakes that if the target doesn't agree you will do something negative in responce. The threat can be as unsubtle as 'Do this or I'll beat you up', to as subtle as, 'If you don't do what I want, I won't be your friend anymore'.
One can imagine an intimidate check being, "Your Grace, if you value our friendship, then you will send your troops to Pickford."
Friends try to intimidate friends in this manner pretty regularly. Sometimes it spells disaster (failed check) and the friendship (or marriage or whatever) is over. Done regularly, and the friendship sours failed check or not. But its a natural human tactic, and we can even imagine honorable heroes like Conan or John Carter making this very sort of claim, "If you have ever valued the friendship of John Carter, then send your warships to the lost sea or else count me no more a friend."... sort of thing.
The fact that the challenge is set up such that you can only get the aid by winning the Duke's trust is itself a railroad. The challenge ought to be 'Get the Duke's Aid'. Winning his trust and support is just one way to do it. Scaring the Duke into going along by repeatedly intimidating him every time he works against you would be another, albiet probably more difficult path (assuming you are less terrifying than Zeus).
Likewise, there are middle grounds. As part of a larger social challenge, dangling carrots via diplomacy and displaying sticks via intimidate is a perfectly acceptable strategy. Maybe the Duke reacts badly to intimidation, but saying that he's immune to it is IMO poor design. Better design might be to say that intimidate has hard difficulty amd each time intimidate is used (successful or not) the DC of all charisma based checks in the challenge goes up by 2.