The situation described above is almost the very definition of a contest and should be between opposing skill checks. In terms of what skill check should be used to oppose intimidation; the DM should decide based on what exactly the other character is doing to resist (which should have been ...
If the players come up with a creative plan that they're invested in and you think is cool, consider whether you're willing to adjust some of what you've prepared (including changing DCs) to help their plan to succeed. Some DMs won't do that, and at some tables 'fudgin...