Another way of keeping your video private is to change in the “Privacy / Settings” the option Only people with a password, to “Hide this video from Vimeo.com” and “Only on sites I choose”.
The reason you might benefit from this is that you have control over the sites that can embed the video. You could even use a normal iframe embed on your app and skip the whole API call if what you want is to show your video on your site and nowhere else. But if you still need to make the call through the API, at least you don’t have the password problem.
This does not answer your question directly, but is an alternative approach to solving the problem.