This is an interesting thread. It sounds like you have a difficult choice, but you're deciding between excellent schools.
I went to Berkeley as an undergrad (I also considered GSPP) and I took quite a few international-oriented classes as a History major. It seemed like GSPP had very few classes for international policy and since there is no International Affairs department at Berkeley you would piece together classes primarily from the Political Science, IAS (International and Area Studies), some ERG, and other departments. My sense of the IAS department when I was there was that it had some amazing professors but was somewhat disorganized as an interdisciplinary department. The political science and ERG departments are exceptional though, and at an info session that GSPP held in November they said they would be introducing an ERG/MPP dual degree soon if you're interested in energy. I don't think it would require a big effort to make your degree more international. I got the sense that it was the norm to take classes in departments around campus. That probably means visiting a professor at office hours to ask if you can join a graduate level seminar.
Berkeley and Stanford are different in a lot of ways, but their graduate departments are top 5 in most areas and to have full access to these classes is really an advantage. Best of luck in your decision.