I would definitely would forget about the cage and drop the rest of your cash on some audio gear like a Zoom H4N or a Zoom H6. You are right that you can get a mic pretty cheap which is fine almost any mic will do. I don't doubt that the internal audio in the gh4 is much better than the gh2 but that still doesn't mean that it is good. It depends really on what you are doing with the camera. If you are just making youtube videos then you don't need great audio but if you are trying to make commercials or doing corporate work then you might want to invest in audio.
