Option 2 is the best, option 1 I am not very convinced about. I don't have experience with Javascript so I'm not sure why you should have to verify that a function exists (option 3).
You can find more information on it here but the reason that you should indeed add a test for this method is to prevent yourself from breaking anything in the future. If you only rely on that one method 5 layers deep in your application, it could be that one day you add code in a higher layer which changes the result but it is not being tested. Or at some level some code has a side-effect which disturbs the result that came from the bowels of your codebase.
Therefore I would suggest you to make a test for each (relevant) level. The question what should I test exactly is probably a little bit preference-oriented but I would argue that the very least you should do is testing whether it returns the correct value, as layed out above.
Should you test that it calls that specific inner method? You could, but that isn't exactly something you do in unit-testing because then you would be testing against the unit's internal workings. You don't care how it works inside, you just care that the function gives you the response that you expected. By coupling these two in your unit-test, you'll end up with a broken test for non-broken code when you decide to refactor the internal workings.