I've never seen that method of integration and I don't believe that's the way you can get your app approved for Instagram.
All I can recommend is the method we suggest in our documentation which is the only known method I'm aware of for getting an app approved. Unfortunately, we don't have any way to speed up the approval process. Months is not normal, but the only recommendation I can make is to double-check that it's been submitted and that all information is correct in the submission.
If you haven't already read over Instagrams approval process workflow, I'd recommend doing that as well in case there's something in there that may help determine the problem:
Thanks,
Alex