Your example is working flawlessly if I just use it as-is in a new html file and go from there, though it is not really usable in exactly that way for us since we want to explicilty avoid to prompt the user for a username and password and are using vuetify, vue-resource and vuex in the project
in the example I gave, you don't prompt the user, the form is already filled with username and password, the example requires you to click a button but that was just a demo, you can easily change that to fill the username and password (you mentioned your app already has the credentials) in hidden inputs, and do the form.submit() on page load, in this way that user does not need to do anything, you will start the flow by redirecting the user to this html page and the page will authenticate without asking anything the user will see just a standard redirect.
i don't have vue experience so i am lost here about how to achieve this with vue related methods but what i am describing is the standard way to achieve the automatic login without prompt from your app, as far i can tell you just need to open a new browser tab in your page that loads this page, this page will do the automatic login without prompt and in the end redirect to studio.