Do I need a vapor control membrane inside the roof if I have a breathable underlay outside?
Vapor barrier against the inside of the building envelope is to prevent interior moisture, caused by normal human activity, from condensing at the dew point. That usually occurs somewhere in the insulation layer and can result in very wet conditions or frost buildup.
So yes, you still want a vapor barrier inside the insulation. What you put on your roof is largely irrelevant.